searchindex.js

Search.setIndex({envversion:42,terms:{polymod:2,finite_r:[2,6,8,3],get_object:8,prefix:6,is_s_integr:5,issquar:3,whose:1,typeerror:[4,5,7,6,1],pth_root:2,swap:1,sped:1,digit:[5,7,1],everi:[7,1,3],vastli:1,lattic:6,ident:[6,1],affect:[5,8],school:3,"__nonzero__":5,z16:8,z10:[6,8],correct:[5,1,8],vector:7,math:[1,8,3],verif:3,initialis:8,erron:8,cunningham_t:3,qsiev:1,naiv:1,ngen:[4,7],direct:[1,8],consequ:8,second:[4,5,6,8],is_rf:4,"10r":7,psu:3,even:[1,3,4,5,6,7,8],neg:[2,5,7,1,8],"new":[1,3],behavior:5,golden_ratio:5,here:[1,3,4,5,7,8],path:[1,3],interpret:[7,1],gens_valu:8,cdef:3,rewrot:1,period_lattic:4,precis:[4,5,7,1,8],galoi:[2,5],fraction_field:7,isomorph:[8,3],residue_field:[4,7],total:5,"byte":1,unit:[4,5,1,8,3],describ:[2,7,3],would:[6,1,8],num_bound:4,has_coerce_map_from:8,call:[1,2,3,4,5,6,7,8],type:[1,2,3,4,5,6,7,8],tell:6,gf2e:6,number_field_bas:4,relat:[4,8,3],notic:7,warn:[2,1,8,3],"__iter__":4,hold:[7,1],gf2x:6,must:[1,2,3,4,5,6,7,8],complex_embed:4,join:[7,8],work:[7,6,1,3],introduc:7,root:[1,2,3,4,5,6,7,8],c140:8,give:[4,5,7,1],sqrt359:3,int_to_z:1,definit:[1,3],fibonacci:5,want:[1,8],david:[2,5,1,8,3],unsign:[5,1],cites:3,is_prime_pow:1,end:[4,7],how:[6,1],int_to_q:5,answer:[1,8,3],verifi:6,neccessarili:4,updat:[8,3],chines:[7,8,3],after:[5,1],befor:1,wrong:[1,3],adic:[1,2,3,4,5,7],demonstr:6,attempt:1,finitefield_pari_ffelt:6,first:[1,3,5,6,7,8],order:[1,2,3,4,5,6,7,8],oper:[7,3],composit:8,is_squar:[1,2,3,5,6,8],over:[8,2,3,4,6,7],becaus:[2,6,1],integer_rational_pow:5,is_perfect_pow:[5,1],"24th":1,baill:1,fit:[2,5,7,1],fix:[1,3,5,6,7,8],better:1,them:[2,1,3],thei:[6,1],squarefree_part:[5,1],prime_to_s_part:5,dickson:3,interrupt:1,choic:[6,8],unpickl:[1,3],luca:3,each:[4,7,1,8,3],finitepolyextel:2,side:3,is_s_unit:5,extract:1,unit_gen:[8,3],content:5,at1:4,uint32:5,log2:1,hang:6,infin:[4,5,7,1,3],free:[5,1],nth:[2,7,1,3],ntl:[2,6],muller:3,convent:4,traceback:[1,2,3,4,5,6,7,8],ist:3,unabl:[4,5,6],inverse_mod:1,factor_cunningham:3,"3908230000000000000000000000000000000000000000000000000000e6":1,rang:[1,3,4,5,7,8],pyrex:7,independ:6,rank:7,alreadi:[1,8],wrapper:1,principalidealdomain:7,mpz_invert:1,too:1,similarli:1,john:[5,1],rational_power_part:5,is_nilpot:3,keyword:[4,5,6,1,8],provid:[7,1,8],"0o12":1,matter:7,equationord:7,"10th":1,mine:1,manner:1,seen:8,absolute_degre:[4,7],seminar:3,ellipticcurv:4,latter:8,simplifi:5,c156:8,though:[4,6,8],object:[7,6,1,8,3],coprime_integ:1,letter:7,cap:[4,7],don:1,complex_numb:1,alarm:1,doe:[5,7,6,1,3],shashank:3,integermod_hom:3,section:[5,3],quadraticfield:5,random:[1,2,4,6,7,8],primes_of_bounded_norm_it:4,mod_ui:5,finite_field_pari_ffelt:6,radic:[5,1],integermodring_gener:[8,3],involv:4,absolut:[4,5,7,1],submit:1,configur:1,siev:1,"__call__":4,likewis:1,ceil:[5,1],congruent:[7,1],cryptographi:3,report:[1,8,3],reconstruct:[1,3],bar:[4,7],"public":3,twice:[1,8,3],result:[1,2,4,5,6,7,8],fail:[1,8],best:1,cofactor:1,crt_basi:7,databas:6,wikipedia:3,gcd_list:1,is_power_of:1,awai:5,pari_ffelt:[2,6],accord:8,never:[7,1],extend:[2,5,7,1,3],xrang:3,maximal_ord:4,extens:[1,2,3,4,5,6,7,8],c22:8,accident:6,local_height_arch:5,c28:8,against:3,compromis:8,kwd:[4,6,1,8],foobar:1,height:[4,5,1],integermod_int64:[6,3],assum:[5,1,8],speak:1,"_pari_":3,nonsens:4,numpi:5,three:3,been:[6,1,8],f31b:8,f31a:8,basic:[7,6],odd_part:1,argument:[4,5,1,8],rational_reconstruct:[1,3],lift:[7,8,3],element_bas:[2,3],save:[1,8],properti:[7,6],mpir:1,aim:6,euclidean:[7,1],calcul:[6,1,3],document:5,long_max:1,zmod:[8,3],n_i:3,sever:[7,6,3],ordinal_str:1,multiplicative_subgroup:8,perform:[8,3],make:[1,3,4,5,6,7],complex:[4,5,7,1],complet:[4,7],ecc2011:3,long_to_z:1,fairli:[7,1],rais:[1,2,3,5,7,8],refin:8,techniqu:3,gp_exp:8,p1363:3,undesir:1,inherit:3,greatest:[1,3],thi:[1,2,3,4,5,6,7,8],left:[2,1],factored_ord:8,just:[2,5,4,1,3],irreducible_el:6,newton:3,ordin:1,yet:[2,6,1,3],previous:8,easi:[7,1],conjug:[2,5,1],els:[4,5,1],formal:1,absolute_norm:5,applic:1,preserv:1,subquoti:8,background:3,lcm:[5,1],lseri:4,c55440:8,specif:3,arbitrari:[4,7,1],manual:1,modulu:[2,6,1,8,3],minpoli:[2,5,3],www:3,right:[7,6,1],nilpot:3,interv:8,maxim:[4,1],intern:[7,6,1,3],indirect:[6,8,3],chaya:3,continued_fract:5,hirzebruch:5,subclass:1,quo_rem:1,defaultdict:7,delecroix:[5,1],polygen:4,is_norm:[5,1],py_new:1,chapter:3,slightli:1,subr:[4,7],"float":[4,5,1],bound:[4,7,1,8],two:[4,7,1,8],down:[2,5,1],wrap:7,suffici:1,support:[1,2,3,5,6,7],avail:4,reli:1,editor:3,fraction:[4,5,7],form:[1,8,3],forc:[8,3],xgcd:[7,1],kroneck:1,taken:5,"0o20":7,"true":[1,2,3,4,5,6,7,8],ieeep1363:3,absent:4,make_integ:1,fundament:1,is_unique_factorization_domain:8,cancel_alarm:1,featur:6,cyclotom:4,exist:[4,7,1,8,3],check:[1,3,4,5,6,7,8],floor:[5,7,1],when:[1,3,4,5,6,7,8],disscrimin:1,test:[1,2,3,4,5,6,7,8],truncate_mod:1,relative_discrimin:4,intent:1,is_integ:[5,7,1],consid:[2,3],uniformli:7,is_subr:[4,7],faster:[5,7,6,1],anywher:1,pseudo:1,ignor:[4,7,6,1],time:[2,6,1,8,3],serious:8,integermod_int:[6,3],global:[5,6,1,8,3],roe:[2,5,1,3],signific:1,"21st":1,depend:[5,1,8,3],decim:5,intermedi:1,primes_first_n:1,finitefieldfactori:6,factor_limit:5,sourc:1,string:[1,2,3,4,5,6,7],asymptot:3,word:[1,3],exact:[5,1],streng:5,foo:6,cunningham:[2,3],"12th":1,iter:[4,6],item:7,postl:3,round:[5,1],upper:4,slower:[6,1,3],sign:[5,1],krull:[7,8],appear:[5,6,1,3],favour:1,uniform:7,current:[2,6,1,8,3],perfect_pow:1,deriv:1,finitefield_pari_ffelt_with_categori:6,coerce_map_from:[5,1],gener:[8,2,3,4,5,6,7],coeffici:[7,1],satisfi:[5,1],nth_root:[2,5,1,3],shift:[1,3],trial:[1,3],behav:[4,8],extrem:1,overrid:3,extra:[4,7,8],modul:[0,7,3],subcategori:8,"1st":1,instal:[1,3],memori:1,prec:[4,5,7,1],live:6,square_root_mod_prime_pow:3,finit:[],discretegaussiandistributionintegersampl:7,logarithm:[5,1],maarten:3,pretend:1,uniqu:[5,7,6,1,8],is_noetherian:[7,8],can:[1,3,4,5,6,7,8],ndigit:1,nearest:5,flajolet:3,scrimshaw:[4,5],cardin:[7,6,8],alwai:[1,3,4,5,6,8],multipl:[1,2,3,5,6,8],pure:5,map:[5,7,8,3],product:[2,1,3],max:[5,3],make_r:5,"4th":1,mai:[1,3,4,5,6,8],is_primitive_root:3,johnson:4,divid:[1,3,4,6,7,8],explicit:6,inform:[4,1,8,3],derickx:3,is_pseudoprime_pow:1,equip:8,still:6,ieee:3,group:[4,6,1,8,3],non:[4,5,7,1,8],initi:[6,1,8,3],half:5,now:1,term:[5,1],name:[4,7,6,8],didn:6,oliv:3,separ:[5,3],hijack:1,domain:[7,1],gaussian:7,contrib:3,coprim:[1,3,4,5,7,8],significantli:1,shown:3,"3rd":1,free_integer_pool:1,rational:1,"100x":1,factori:[6,1,8],bnfisnorm:5,runtimeerror:3,org:3,ord:[5,1],care:[1,8],multiplicative_gener:[8,3],wai:[7,6,1,8],prescrib:8,where:[1,2,3,5,7,8],zech:6,thing:[1,3],place:[4,5],test_bit:1,think:6,lambda:7,origin:1,directli:1,generalised_log:3,"long":[1,2,3,4,5,6,7,8],bradshaw:[7,6,1,8,3],lexicograph:6,size:[2,6,1,8,3],given:[1,2,3,5,6,7,8],workaround:[5,1],forcabl:1,euclideandomainel:1,numberfield:[4,5,1],draft:3,conveni:1,arith:[1,3],selmer_group_iter:4,specifi:[1,3,4,6,7,8],domin:1,than:[1,3,4,5,6,7,8],archimedean:5,posit:[4,5,7,1,8],seri:1,is_on:[5,1,3],craig:[5,3],ani:[4,7,6,1,3],"13th":1,notimplementederror:[2,5,7,3],squar:[2,5,1,8,3],moreov:8,note:[1,2,3,5,6,7,8],ideal:[1,2,3,4,7,8],denomin:[4,5,1],an_el:3,take:[5,1,3],lift_map:7,begin:[5,1],sure:[5,7,1,8,3],trace:[2,5,3],sagemath:3,"1rl":1,beta:[5,1],pair:[4,5,7,1],unramifi:4,synonym:8,phin:3,later:8,runtim:[5,1,8],show:[4,3],permiss:6,v_p:1,codomain:3,slot:1,onli:[1,2,3,4,5,6,7,8],dict:1,variou:[4,7,6,8,3],get:[7,6,1,3],repr:6,cannot:[5,7,1],"import":[1,2,3,4,5,6,7,8],gen:[4,7,6,8,3],requir:[5,6,1],prime:[1,2,3,4,5,6,7,8],yield:[4,6,3],pynac:1,summari:3,loeffler:1,is_absolut:4,infinit:[4,5,7,1],timeit:1,number_field:4,monoid:8,enumer:[7,6,1,8],label:6,enough:[1,3],between:[5,7,1,8],harvei:[5,1],across:6,parent:[1,2,3,4,5,6,7,8],integermodr:[8,3],nile:4,improv:[5,1],among:1,pow:8,uncondition:5,balanced_sum:1,period:5,baltimor:[2,3],pol:[5,1],poli:[4,7,8],didier:[1,3],zimmermann:3,invert:[8,3],algebraic_closur:4,invers:[5,7,1,3],nextprim:1,valueerror:[1,2,3,4,5,6,7,8],doesn:[5,7,1],c3188646:8,those:4,"case":[1,2,3,4,5,6,7,8],euler_phi:3,siguna:3,henc:[1,8],uncov:8,is_integrally_clos:7,create_key_and_extra_arg:[6,8],"__init__":1,develop:8,author:[1,2,3,4,5,6,7,8],integermod_gmp:[6,3],same:[1,3,4,6,7,8],binari:1,html:3,samp:7,integer_to_integermod:3,mani:[7,1],defn:[4,1],appropri:2,rationalfield:[4,3],without:[5,8,3],summer:3,pari_mod:6,ramanna:3,polish:3,speed:1,is_pseudoprim:1,powermodm_ui:1,except:[2,1,3],littl:[7,1],exercis:1,is_finitefieldel:2,real:[4,5,7,1],around:[7,1],vivek:3,finite_field_ntl_gf2:6,psw:1,mod:[7,1,8,3],integ:[],either:[5,7,6,1,8],output:[1,2,3,4,5,6,7,8],"111th":1,square_root:[2,3],magma:[6,1],local_height:5,nonzero:[5,1,3],is_integermod:3,"23rd":1,base_r:6,refer:[5,7,3],power:[1,2,3,4,5,6,8],ration:[],is_rat:5,greatli:5,joyner:[8,3],acm:[2,3],degre:[4,7,6,1,8],johnston:[2,3],rrrt:3,morphism:[4,5,1,3],routin:2,effici:3,quotient_r:8,strip:3,your:8,log:[2,5,6,1,3],aren:1,omega:4,brute:3,start:[4,7,6,1],again:4,sagesagesag:1,lot:1,strictli:[4,1],previous_prime_pow:1,minimal_polynomi:[2,3],tupl:[4,5,7,1,8],pull:1,tripl:1,possibl:[4,6,1,8,3],"default":[1,2,3,4,5,6,7,8],embed:[4,7,6,8],creat:[1,3,4,6,7,8],tornaria:[5,1],decreas:8,proport:7,incorrect:[1,3],denot:[7,3],get_data:1,qqbar:5,field:[],binomi:1,valid:7,denom:5,is_unit:[1,3],architectur:3,integraldomain:8,annex:3,sequenc:8,symbol:[5,1,3],is_prime_field:[4,8],docstr:[4,1],polynomi:[1,2,3,4,5,6,7,8],pool:1,quadratic_nonresidu:8,reduc:[2,7,1,3],square_root_mod_prim:3,elem_cach:6,unset:1,represent:[4,5,6,1],all:[1,2,3,4,5,6,7,8],illustr:[7,6,1,8,3],int_fast32_t:3,deprecationwarn:3,abil:1,follow:[1,2,3,4,6,8],edg:5,albrecht:[6,1],articl:3,neglig:7,norm:[2,5,1,3],liter:1,krull_dimens:[7,8],fals:[1,2,3,4,5,6,7,8],morain:3,fall:3,veri:[2,6,1,8,3],ticket:[1,3,4,5,6,7,8],list:[1,2,3,4,5,7,8],arithmeticerror:[2,5,1,3],small:[1,2,3,5,6,7],fast_method:4,"20th":1,dimens:[7,8],zero:[5,7,1,3],design:3,pass:[4,7,6,1,3],further:[7,6,1],integermodfactori:8,what:8,sum:[2,5,1],abl:4,version:[5,6,1,8,3],is_irreduc:1,method:[1,3,4,5,6,7],contrast:8,full:[4,5],hash:[5,1],behaviour:1,modular:3,strong:1,valu:[1,3,4,5,6,7],trunc:5,search:0,divisor:[2,1,8,3],popcount:1,"__xor__":3,amount:1,doctest:[5,6,1,8,3],bezout:1,exact_flag:1,narrow:1,magnitud:1,quotient:[5,7,1,8,3],via:[6,1,3],shorthand:1,primit:[2,7,6,1],is_primefinitefield:6,deprec:[1,3],coercion:[4,7,6,1,8],convert_map_from:[5,3],hexadecim:[7,1],proceed:[2,3],distinct:[4,6],is_integr:[5,1],selmer:4,change_variable_nam:6,coverag:[4,5],almost:5,binaryrecurrencesequ:3,nativeintstruct:3,more:[1,3,4,5,6,7,8],desir:[5,7,1],canon:8,hundr:3,probabilist:[1,8,3],flag:[4,6,1],stick:1,particular:[6,8],known:1,cach:[2,6,8,3],none:[1,2,3,4,5,6,7,8],endpoint:7,def:8,bogu:8,"_bnfisnorm":5,accept:[5,7,6,1],minimum:1,slow_luca:3,cours:[1,8],prod_:3,rather:[1,3],anoth:6,isqrt:1,divis:[1,3,4,5,7,8],deshomm:[1,3],prime_divisor:1,isn:1,atkin92probabilist:3,algebra:[4,1,8,3],"0123456789abcdef":1,reflect:1,associ:[4,1],d13:3,caus:1,zerodivisionerror:[7,1,3],"_xgcd":1,paper:[2,3],through:[4,7,1],paramet:[5,7,6,8],hendrik:1,power_basi:4,jacobi:[1,3],might:[1,8,3],fool:6,good:[2,1],"return":[1,2,3,4,5,6,7,8],lowercas:3,inria:3,bigger:3,eventu:1,easili:[6,1],iff:[7,3],found:[1,8,3],unicod:1,truncat:[7,1],valuat:[4,5,7,1,3],memoryerror:1,finitefield_prime_modn_with_categori:6,functor:4,realli:[5,6,1],principal_id:8,expect:[5,7,1],hart:1,reduct:7,znlog:3,robert:[7,6,1,8,3],research:3,is_field:[4,7,6,8],print:[7,6,1,3],den_bound:4,multifactori:1,advanc:6,like:[4,8],reason:[1,3],base:[],put:8,basi:[4,7],perhap:8,previous_prim:1,assign:8,singleton:4,misc:4,number:[],done:[4,1,3],least:[5,7,6,1],pablo:[5,1],mistakenli:8,differ:[7,1,8,3],exponenti:1,interact:4,construct:[4,7,6,1,8],store:3,option:[1,2,3,4,5,6,7,8],relationship:1,pari:[1,2,3,4,5,6,8],part:[5,1,8],c62500:8,std:5,king:[8,3],cyclic:[1,8,3],sqrtrem:1,remov:[4,5],reus:4,str:[5,7,1],toward:[5,1],naqi:[5,1],randomli:4,comput:[1,2,3,4,5,6,7,8],packag:[1,3],padto:1,equival:3,randint:3,self:[1,2,3,4,5,8],also:[1,3,4,5,6,7,8],lucas_number2:3,distribut:[4,7],previou:[4,1],most:[1,2,3,4,5,6,7,8],alpha:[2,6],compar:1,clear:3,exact_log:1,exp:[4,1],usual:[6,1,3],ffelt:6,finitefield_ntl_gf2e_with_categori:6,range_by_height:4,probabilitst:8,session:8,particularli:1,worri:6,find:[5,1,3],impact:8,access:1,coerc:[7,1],solut:8,factor:[1,2,3,5,6,8],abelian:8,express:1,nativ:[6,1,3],commutativ:8,fastest:1,givaro:[2,6],mpz_rrandomb:7,restart:8,finitefield_givaro:6,crt:[2,7,1,8,3],whenev:1,is_pow:1,common:[1,3],vincent:[5,1],class_numb:[4,1],finitefield_ntl_gf2:6,set:[1,3,4,5,7,8],dump:[4,5,1,3],cartesian_product_iter:3,see:[1,2,3,4,5,6,7,8],arg:[4,6,1],close:[7,1],"11th":1,someth:1,f23:8,won:5,altern:[1,3],signatur:4,selmer_group:4,numer:[4,5,7,1],complement:1,"10000000000000000000001r":1,distinguish:8,both:4,last:[1,2,3,4,5,6,7,8],annual:[2,3],real_mpfr:1,whole:8,load:[4,5,1,8,3],simpli:[4,5,7,1],undefin:1,point:[5,1],residu:[4,7,8,3],mistak:8,zeta:[4,7],backend:1,symposium:[2,3],fractionfield:4,empti:[4,1,3],c16777216:8,imag:[5,1],convers:[4,5,1],gmp:[5,7,1],bill:1,"while":[4,6],abov:[5,7,1,8],error:[5,7,1,3],loop:1,propag:1,vol:3,itself:[5,1,3],costli:8,quadrat:[1,8,3],incompat:3,minim:[2,5,6,1,3],supercalifragilisticexpialidoci:6,belong:8,next_probable_prim:1,u_p:1,octal:[7,1],continued_fraction_list:5,higher:3,optim:[5,7,1],all_complex:4,user:[6,8],provabl:[6,1],typic:3,recent:[1,2,3,4,5,6,7,8],older:6,entri:[5,7,1],expens:3,runtimewarn:1,venkatesh:3,explan:7,complexnumb:1,theoret:3,theorem:[7,8,3],input:[1,2,3,4,5,6,7,8],subsequ:7,marco:5,march:3,stein:[5,6,1,8,3],format:1,big:1,subfield:2,kash:1,bit:[4,5,7,1,3],characterist:[8,2,3,4,5,6,7],print_mod:[4,7],is_integermodr:8,resolv:3,collect:[4,7,1],princip:7,often:[5,1],creation:[7,6],some:[1,3,4,5,6,7],back:3,understood:7,unspecifi:1,sampl:7,pep:1,per:1,mathemat:3,larg:[1,3],next_prime_pow:1,prod:[1,3],frobeniu:2,anna:[2,3],run:[2,7,6,8],step:[5,7,1],cyclotomicfield:4,integerwrapp:1,prove:1,integermod_abstract:3,gamma:[5,1],realfield:[4,5,1],bailli:1,primarili:1,ensur:1,kohel:3,inclus:[4,7],fast:[5,6,1,3],custom:1,arithmet:[5,7,6,1,8],includ:[5,7,1,8],int_:1,polmod:6,properli:[4,6],repeatedli:7,line:3,int8:5,consist:[4,5,6,8,3],caller:1,similar:[1,3],impl:[2,6],curv:[4,1],nbit:1,constant:3,global_height_non_arch:5,euclidean_degre:1,isprim:1,repres:[1,2,3,4,6,7],incomplet:1,invalid:1,additive_ord:[2,5,1,3],quotientring_gener:8,grh:5,nice:4,create_object:[6,8],william:[5,6,1,8,3],algorithm:[1,2,3,5,6,7,8],monic:[6,1],notion:1,discrimin:[4,1],code:[5,7,1,8,3],partial:[5,7],is_subcategori:8,finite_field_prime_modn:6,queri:[7,6],"113th":1,ellipt:[4,1],edu:3,global_height:[5,1],cython:[1,8],simon:[8,3],sens:[7,3],factorint:3,citro:[5,3],alphanumer:7,sage:[1,2,3,4,5,6,7,8],uniquefactori:[6,8],volum:3,joel:1,"try":[5,6,1,3],rational_field:4,mpz_t:[7,1,3],pleas:[1,8],impli:7,smaller:6,factored_unit_ord:8,natur:[5,7,1],"11a":4,odd:[5,1,8,3],compat:[4,5,6,1,3],index:[0,1,3],polynomial_zmod_flint:[6,3],integermod:[2,8,3],centerlift:3,is_finit:[4,7,8],squarefre:1,len:[7,1,8,3],closur:4,let:[7,3],sinc:[1,3,4,5,6,7,8],convert:[1,3,4,5,6,8],powermod:1,larger:[4,6,1],is_nth_pow:5,rdf:[4,5,1],chang:[5,6,1,8],danger:6,appli:5,approxim:[5,1],"boolean":[5,7,1],from:[1,2,3,4,5,6,7,8],unit_group_expon:8,doubl:4,next:[2,7,1,8],few:[7,1],lcm_list:1,finitefield_givaro_with_categori:6,remaind:[5,7,1,8,3],sort:[5,7,1],sriniva:3,trail:1,cputim:1,alia:[5,6],integer_mod:[6,3],endian:[7,1],proof:[1,3,4,5,6,7,8],process:1,precompute_t:3,high:1,gcd:[5,7,1,3],occur:[7,3],instead:[1,2,3,5,6,7],ffinit:6,sin:6,int16:5,z_to_q:5,multiplicative_group_is_cycl:8,relative_norm:5,gonzalo:[5,1],unit_group:8,alloc:1,"22nd":1,correspond:[2,8,4],element:[],issu:[1,3],allow:[7,1,8],move:3,is_rationalfield:4,multiplicative_ord:[1,2,3,5,6,8],perfect:[2,5,1,3],integermod_to_integermod:3,chosen:8,lucas_sequ:3,crash:[5,6],greater:[7,1,3],handl:[1,3],extra_arg:8,facilit:1,trac:[1,3,4,5,6,7,8],anyth:7,edit:3,mode:5,modn:6,"static":1,frac:4,meth:2,patch:1,tenth:[2,3],special:1,out:[2,1,8],variabl:[2,7,1],matrix:2,outlier:7,categori:[1,3,5,6,7,8],rel:[4,7,1,3],rem:1,leverag:1,matric:7,reduction_map:7,random_el:[1,2,3,4,7,8],ecm:1,dictionari:1,prime_rang:3,quo:[7,3],c110:8,"1001st":1,napoli:[5,1],cremona:[5,1],could:3,keep:1,length:[5,1],enforc:4,outsid:4,rishikesh:1,divide_knowing_divisible_bi:1,lenstra:1,jafferi:[5,1],unit_group_ord:8,global_height_arch:5,finitefield:[2,6,3],unknown:8,system:1,messag:8,interfac:1,siam:[2,3],travi:[4,5],discrete_log:3,exactli:8,alarminterrupt:1,structur:[2,5,6,1,8],trial_divis:1,f19:8,"0b101010":1,prime_to_m_part:1,favorit:1,atkin:3,c20499647385305088000000:8,have:[1,4,5,6,7,8],need:[1,3,5,6,7,8],optimis:1,min:3,mix:1,builtin:5,discret:[2,7,3],which:[1,2,3,4,5,6,7,8],singh:3,singl:[4,7],unless:[6,1],realnumb:1,integer_mod_r:[8,3],histogram:7,"class":[],pushout:6,request:3,univari:[8,3],determin:[4,5,1,3],fact:[7,1,8],precprim:1,text:3,verbos:1,trivial:[4,8],homomorph:[4,3],square_roots_of_on:8,should:[4,1],smallest:[4,5,1,3],"112th":1,local:5,c2058:8,next_prim:[4,6,1,8,3],notat:1,familiar:1,increas:1,integr:[5,7,1,8],contain:[4,7,6,1,8],moduli:[8,3],modulo:[],charpoli:[2,5,6,3],is_integral_domain:8,endomorph:4,int_to_integermod:3,qth:[2,3],conjectur:1,mainli:3,tend:1,list_of_elements_of_multiplicative_group:8,written:5,integermod_to_integ:3,kei:[6,8,3],entir:1,otherwis:[1,2,3,5,6,7,8],addit:[1,2,3,4,5,6,7],equal:[4,5,6,1],etc:7,instanc:[4,7,6,8],equat:7,x_i:3,is_prim:[1,8],tonelli:3,respect:[5,1,8,3],inverse_of_unit:1,compon:[5,8],treat:[4,1],immedi:1,check_irreduc:6,assert:[2,1,8],finiteringel:[2,3],determinist:8,multi:1,prime_factor:1,defin:[1,2,3,4,5,6,7],is_padic_squar:5,makenativeintstruct:3,howev:[6,1,8,3],srang:[5,1],irreduc:[6,1],noetherian:[7,8],complexfield:1,is_squarefre:1,uniti:[4,7],sqrt:[2,5,1,3],member:1,python:[4,5,7,1,8],largest:[1,3],"0x1a":7,fieldel:5,http:3,cubic:2,expans:5,upon:5,effect:[8,3],dealloc:1,expand:5,off:[1,3],center:7,well:[7,1],"0x12":1,"0x10":7,exampl:[1,2,3,4,5,6,7,8],command:3,choos:[4,1],powb:8,powa:8,subgroup:[4,8],less:[4,1,8,3],expected_valu:5,obtain:[8,3],mistaken:8,expon:[5,1],mohler:1,add:1,c18:8,somindu:3,c12:8,semigroup:8,c10:8,c16:8,int32:5,know:[5,6,1,8],"2nd":[5,1],recurs:[1,3],lucas_q1:3,arbitrarili:3,sagex:8,necessari:[2,5,1,3],martin:[6,1],page:0,revert:8,val_unit:[5,1],guarante:[6,1],absolute_discrimin:4,librari:[5,6,1],lead:6,discrete_gaussian_integ:7,leak:1,avoid:[6,8],"0th":1,"14th":1,commutativeringel:2,leav:6,duplic:4,pseudoprim:[6,1],imaginari:[5,1],although:1,about:[7,6,1],actual:[7,6,1],finite_field_givaro:6,testsuit:[2,7,6,8],constructor:[6,1,3],q_to_z:5,primal:[1,3,4,5,6,7,8],ring_of_integ:4,automat:[7,8],conwai:6,merg:1,pos88:3,much:1,"var":[2,5,6,3],overflowerror:1,"function":[1,2,3,4,5,7,8],shank:3,continu:[2,5,4,3],bug:[5,1,3],count:1,whether:[5,7,1,8,3],wish:6,below:[7,1,3],limit:[5,6,1],unnot:1,problem:[2,1],evalu:[5,3],"int":[1,2,3,4,5,6,7,8],dure:8,implement:[2,5,6,1,3],pth_power:2,probabl:[7,1],integerring_class:[4,7],detail:[4,7,1,8,3],other:[1,3,4,5,6,7,8],bool:[1,2,3,5,6,8],branch:1,int_fast64_t:3,float16:5,stat:7,repeat:5,jung:5,trailing_zero_bit:1,integer_r:[4,7],you:[1,3,4,5,6,7,8]},objtypes:{"0":"py:module","1":"py:method","2":"py:class","3":"py:function"},objnames:{"0":["py","module","Python module"],"1":["py","method","Python method"],"2":["py","class","Python class"],"3":["py","function","Python function"]},filenames:["index","sage/rings/integer","sage/rings/finite_rings/element_base","sage/rings/finite_rings/integer_mod","sage/rings/rational_field","sage/rings/rational","sage/rings/finite_rings/constructor","sage/rings/integer_ring","sage/rings/finite_rings/integer_mod_ring"],titles:["Standard Commutative Rings","Elements of the ring <span class=\"math\">\\(\\ZZ\\)</span> of integers","Base class for finite field elements","Elements of <span class=\"math\">\\(\\ZZ/n\\ZZ\\)</span>","Field <span class=\"math\">\\(\\QQ\\)</span> of Rational Numbers","Rational Numbers","Finite Fields","Ring <span class=\"math\">\\(\\ZZ\\)</span> of Integers","Ring <span class=\"math\">\\(\\ZZ/n\\ZZ\\)</span> of integers modulo <span class=\"math\">\\(n\\)</span>"],objects:{"sage.rings.finite_rings":{integer_mod:[3,0,0,"-"],element_base:[2,0,0,"-"],integer_mod_ring:[8,0,0,"-"],constructor:[6,0,0,"-"]},"sage.rings.rational_field.RationalField":{relative_discriminant:[4,1,1,""],range_by_height:[4,1,1,""],power_basis:[4,1,1,""],absolute_degree:[4,1,1,""],algebraic_closure:[4,1,1,""],selmer_group:[4,1,1,""],zeta:[4,1,1,""],class_number:[4,1,1,""],completion:[4,1,1,""],complex_embedding:[4,1,1,""],number_field:[4,1,1,""],primes_of_bounded_norm_iter:[4,1,1,""],maximal_order:[4,1,1,""],is_field:[4,1,1,""],construction:[4,1,1,""],is_finite:[4,1,1,""],embeddings:[4,1,1,""],degree:[4,1,1,""],is_prime_field:[4,1,1,""],absolute_discriminant:[4,1,1,""],random_element:[4,1,1,""],is_subring:[4,1,1,""],gen:[4,1,1,""],selmer_group_iterator:[4,1,1,""],extension:[4,1,1,""],places:[4,1,1,""],characteristic:[4,1,1,""],discriminant:[4,1,1,""],gens:[4,1,1,""],ngens:[4,1,1,""],residue_field:[4,1,1,""],signature:[4,1,1,""],order:[4,1,1,""],is_absolute:[4,1,1,""]},"sage.rings.finite_rings.integer_mod_ring.IntegerModRing_generic":{krull_dimension:[8,1,1,""],is_integral_domain:[8,1,1,""],square_roots_of_one:[8,1,1,""],quadratic_nonresidue:[8,1,1,""],cardinality:[8,1,1,""],unit_group_exponent:[8,1,1,""],list_of_elements_of_multiplicative_group:[8,1,1,""],is_field:[8,1,1,""],modulus:[8,1,1,""],factored_unit_order:[8,1,1,""],is_finite:[8,1,1,""],degree:[8,1,1,""],multiplicative_subgroups:[8,1,1,""],is_prime_field:[8,1,1,""],random_element:[8,1,1,""],unit_group:[8,1,1,""],unit_gens:[8,1,1,""],is_unique_factorization_domain:[8,1,1,""],unit_group_order:[8,1,1,""],multiplicative_generator:[8,1,1,""],extension:[8,1,1,""],characteristic:[8,1,1,""],multiplicative_group_is_cyclic:[8,1,1,""],order:[8,1,1,""],is_noetherian:[8,1,1,""],field:[8,1,1,""],factored_order:[8,1,1,""]},"sage.rings.rational.Z_to_Q":{section:[5,1,1,""]},"sage.rings.rational":{Z_to_Q:[5,2,1,""],Rational:[5,2,1,""],int_to_Q:[5,2,1,""],is_Rational:[5,3,1,""],make_rational:[5,3,1,""],Q_to_Z:[5,2,1,""],rational_power_parts:[5,3,1,""],integer_rational_power:[5,3,1,""]},"sage.rings.finite_rings.integer_mod.IntegerMod_abstract":{square_root:[3,1,1,""],is_nilpotent:[3,1,1,""],is_square:[3,1,1,""],generalised_log:[3,1,1,""],rational_reconstruction:[3,1,1,""],charpoly:[3,1,1,""],multiplicative_order:[3,1,1,""],polynomial:[3,1,1,""],nth_root:[3,1,1,""],valuation:[3,1,1,""],sqrt:[3,1,1,""],is_unit:[3,1,1,""],modulus:[3,1,1,""],norm:[3,1,1,""],centerlift:[3,1,1,""],trace:[3,1,1,""],additive_order:[3,1,1,""],is_primitive_root:[3,1,1,""],log:[3,1,1,""],is_one:[3,1,1,""],minimal_polynomial:[3,1,1,""],crt:[3,1,1,""],minpoly:[3,1,1,""]},"sage.rings.finite_rings.constructor":{is_PrimeFiniteField:[6,3,1,""],FiniteFieldFactory:[6,2,1,""]},"sage.rings.rational.Rational":{global_height_non_arch:[5,1,1,""],numerator:[5,1,1,""],is_square:[5,1,1,""],continued_fraction:[5,1,1,""],period:[5,1,1,""],sign:[5,1,1,""],mod_ui:[5,1,1,""],charpoly:[5,1,1,""],conjugate:[5,1,1,""],local_height:[5,1,1,""],global_height:[5,1,1,""],is_integer:[5,1,1,""],is_S_unit:[5,1,1,""],is_integral:[5,1,1,""],nth_root:[5,1,1,""],global_height_arch:[5,1,1,""],floor:[5,1,1,""],is_nth_power:[5,1,1,""],support:[5,1,1,""],is_perfect_power:[5,1,1,""],sqrt:[5,1,1,""],content:[5,1,1,""],numer:[5,1,1,""],factor:[5,1,1,""],denom:[5,1,1,""],continued_fraction_list:[5,1,1,""],absolute_norm:[5,1,1,""],imag:[5,1,1,""],real:[5,1,1,""],val_unit:[5,1,1,""],is_S_integral:[5,1,1,""],trace:[5,1,1,""],relative_norm:[5,1,1,""],squarefree_part:[5,1,1,""],denominator:[5,1,1,""],additive_order:[5,1,1,""],multiplicative_order:[5,1,1,""],is_norm:[5,1,1,""],ord:[5,1,1,""],is_padic_square:[5,1,1,""],valuation:[5,1,1,""],trunc:[5,1,1,""],is_one:[5,1,1,""],local_height_arch:[5,1,1,""],ceil:[5,1,1,""],list:[5,1,1,""],height:[5,1,1,""],prime_to_S_part:[5,1,1,""],norm:[5,1,1,""],str:[5,1,1,""],minpoly:[5,1,1,""],round:[5,1,1,""],gamma:[5,1,1,""]},"sage.rings.finite_rings.integer_mod.Integer_to_IntegerMod":{section:[3,1,1,""]},"sage.rings.finite_rings.constructor.FiniteFieldFactory":{create_key_and_extra_args:[6,1,1,""],create_object:[6,1,1,""]},"sage.rings.finite_rings.integer_mod.IntegerMod_int":{is_unit:[3,1,1,""],lift:[3,1,1,""],gcd:[3,1,1,""],sqrt:[3,1,1,""],is_one:[3,1,1,""]},"sage.rings.finite_rings.integer_mod.IntegerMod_gmp":{is_unit:[3,1,1,""],lift:[3,1,1,""],gcd:[3,1,1,""],is_one:[3,1,1,""]},"sage.rings.finite_rings.integer_mod":{IntegerMod:[3,3,1,""],IntegerMod_to_IntegerMod:[3,2,1,""],IntegerMod_to_Integer:[3,2,1,""],lucas:[3,3,1,""],Int_to_IntegerMod:[3,2,1,""],is_IntegerMod:[3,3,1,""],Integer_to_IntegerMod:[3,2,1,""],square_root_mod_prime_power:[3,3,1,""],IntegerMod_gmp:[3,2,1,""],IntegerMod_int64:[3,2,1,""],lucas_q1:[3,3,1,""],IntegerMod_abstract:[3,2,1,""],IntegerMod_hom:[3,2,1,""],NativeIntStruct:[3,2,1,""],slow_lucas:[3,3,1,""],IntegerMod_int:[3,2,1,""],Mod:[3,3,1,""],square_root_mod_prime:[3,3,1,""],makeNativeIntStruct:[3,3,1,""],mod:[3,3,1,""]},"sage.rings.integer":{int_to_Z:[1,2,1,""],Integer:[1,2,1,""],free_integer_pool:[1,3,1,""],long_to_Z:[1,2,1,""],IntegerWrapper:[1,2,1,""],LCM_list:[1,3,1,""],GCD_list:[1,3,1,""],make_integer:[1,3,1,""],is_Integer:[1,3,1,""]},"sage.rings.integer_ring.IntegerRing_class":{krull_dimension:[7,1,1,""],residue_field:[7,1,1,""],absolute_degree:[7,1,1,""],is_integrally_closed:[7,1,1,""],zeta:[7,1,1,""],completion:[7,1,1,""],is_field:[7,1,1,""],parameter:[7,1,1,""],degree:[7,1,1,""],is_finite:[7,1,1,""],random_element:[7,1,1,""],is_subring:[7,1,1,""],gen:[7,1,1,""],extension:[7,1,1,""],characteristic:[7,1,1,""],gens:[7,1,1,""],ngens:[7,1,1,""],range:[7,1,1,""],is_noetherian:[7,1,1,""],fraction_field:[7,1,1,""],order:[7,1,1,""],quotient:[7,1,1,""]},"sage.rings.finite_rings.integer_mod_ring.IntegerModFactory":{get_object:[8,1,1,""],create_key_and_extra_args:[8,1,1,""],create_object:[8,1,1,""]},"sage.rings.finite_rings.integer_mod.NativeIntStruct":{precompute_table:[3,1,1,""]},"sage.rings.finite_rings.element_base.FinitePolyExtElement":{square_root:[2,1,1,""],minimal_polynomial:[2,1,1,""],nth_root:[2,1,1,""],frobenius:[2,1,1,""],minpoly:[2,1,1,""],is_square:[2,1,1,""],sqrt:[2,1,1,""],additive_order:[2,1,1,""],pth_root:[2,1,1,""],pth_power:[2,1,1,""],charpoly:[2,1,1,""],multiplicative_order:[2,1,1,""],trace:[2,1,1,""],norm:[2,1,1,""]},"sage.rings.rational_field":{is_RationalField:[4,3,1,""],frac:[4,3,1,""],RationalField:[4,2,1,""]},"sage.rings":{integer:[1,0,0,"-"],rational_field:[4,0,0,"-"],rational:[5,0,0,"-"],integer_ring:[7,0,0,"-"]},"sage.rings.finite_rings.integer_mod_ring":{IntegerModFactory:[8,2,1,""],IntegerModRing_generic:[8,2,1,""],crt:[8,3,1,""],is_IntegerModRing:[8,3,1,""]},"sage.rings.integer.Integer":{real:[1,1,1,""],trailing_zero_bits:[1,1,1,""],is_prime_power:[1,1,1,""],multifactorial:[1,1,1,""],next_prime:[1,1,1,""],is_irreducible:[1,1,1,""],is_square:[1,1,1,""],global_height:[1,1,1,""],sign:[1,1,1,""],previous_prime_power:[1,1,1,""],previous_prime:[1,1,1,""],list:[1,1,1,""],is_pseudoprime_power:[1,1,1,""],rational_reconstruction:[1,1,1,""],multiplicative_order:[1,1,1,""],factor:[1,1,1,""],additive_order:[1,1,1,""],divide_knowing_divisible_by:[1,1,1,""],ord:[1,1,1,""],inverse_of_unit:[1,1,1,""],ordinal_str:[1,1,1,""],binary:[1,1,1,""],nbits:[1,1,1,""],is_integral:[1,1,1,""],nth_root:[1,1,1,""],floor:[1,1,1,""],gcd:[1,1,1,""],support:[1,1,1,""],is_perfect_power:[1,1,1,""],sqrt:[1,1,1,""],prime_factors:[1,1,1,""],is_power_of:[1,1,1,""],euclidean_degree:[1,1,1,""],class_number:[1,1,1,""],kronecker:[1,1,1,""],jacobi:[1,1,1,""],is_unit:[1,1,1,""],quo_rem:[1,1,1,""],bits:[1,1,1,""],imag:[1,1,1,""],trial_division:[1,1,1,""],val_unit:[1,1,1,""],ndigits:[1,1,1,""],squarefree_part:[1,1,1,""],exp:[1,1,1,""],factorial:[1,1,1,""],next_prime_power:[1,1,1,""],xgcd:[1,1,1,""],denominator:[1,1,1,""],odd_part:[1,1,1,""],ceil:[1,1,1,""],is_pseudoprime:[1,1,1,""],popcount:[1,1,1,""],binomial:[1,1,1,""],conjugate:[1,1,1,""],perfect_power:[1,1,1,""],radical:[1,1,1,""],prime_divisors:[1,1,1,""],coprime_integers:[1,1,1,""],valuation:[1,1,1,""],divisors:[1,1,1,""],log:[1,1,1,""],digits:[1,1,1,""],next_probable_prime:[1,1,1,""],is_one:[1,1,1,""],is_integer:[1,1,1,""],powermod:[1,1,1,""],crt:[1,1,1,""],is_squarefree:[1,1,1,""],is_prime:[1,1,1,""],sqrtrem:[1,1,1,""],powermodm_ui:[1,1,1,""],numerator:[1,1,1,""],is_norm:[1,1,1,""],divides:[1,1,1,""],inverse_mod:[1,1,1,""],exact_log:[1,1,1,""],str:[1,1,1,""],prime_to_m_part:[1,1,1,""],test_bit:[1,1,1,""],isqrt:[1,1,1,""],gamma:[1,1,1,""]},"sage.rings.integer_ring":{IntegerRing:[7,3,1,""],IntegerRing_class:[7,2,1,""],crt_basis:[7,3,1,""],is_IntegerRing:[7,3,1,""]},"sage.rings.finite_rings.integer_mod.IntegerMod_int64":{is_unit:[3,1,1,""],lift:[3,1,1,""],gcd:[3,1,1,""],is_one:[3,1,1,""]},"sage.rings.finite_rings.element_base":{FiniteRingElement:[2,2,1,""],FinitePolyExtElement:[2,2,1,""],is_FiniteFieldElement:[2,3,1,""]},"sage.rings.rational.Q_to_Z":{section:[5,1,1,""]}},titleterms:{modulo:8,"class":2,ration:[4,5],finit:[2,6],indic:0,standard:0,field:[2,6,4],base:2,integ:[7,1,8],tabl:0,number:[4,5],ring:[0,7,1,8],element:[2,1,3],commut:0}})