4) Steinberg vs Multiplicative Reduction.sagews

nf = ModularForms(Gamma0(50),2,prec=20).newforms()
nf[0]
print"\n"
nf[1]

#nf[0] and nf[1] are quadratic twists of each other, by the non-trivial character of Gal(Q(sqrt(5))/Q)

q - q^2 + q^3 + q^4 - q^6 + 2*q^7 - q^8 - 2*q^9 - 3*q^11 + q^12 - 4*q^13 - 2*q^14 + q^16 - 3*q^17 + 2*q^18 + 5*q^19 + O(q^20) q + q^2 - q^3 + q^4 - q^6 - 2*q^7 + q^8 - 2*q^9 - 3*q^11 - q^12 + 4*q^13 - 2*q^14 + q^16 + 3*q^17 - 2*q^18 + 5*q^19 + O(q^20)

#Take the corresponding quadratic twist of E
E = EllipticCurve([1,0,1,-1,-2])
E5 = E.quadratic_twist(5)
E5

Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 13*x - 219 over Rational Field

E.local_data(2)

print "\n"
print "\n"

E5.local_data(2)

Local data at Principal ideal (2) of Integer Ring: Reduction type: bad non-split multiplicative Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 - x - 2 over Rational Field Minimal discriminant valuation: 1 Conductor exponent: 1 Kodaira Symbol: I1 Tamagawa Number: 1 Local data at Principal ideal (2) of Integer Ring: Reduction type: bad split multiplicative Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 13*x - 219 over Rational Field Minimal discriminant valuation: 1 Conductor exponent: 1 Kodaira Symbol: I1 Tamagawa Number: 1