# Revision history [back]

### Reductions of elliptic curves over number fields

I have found two elliptic curves over a number field whose reductions are isomorphic to supersingular elliptic curves (used Deuring Lifting Theorem). How can I check which reduction is isomorphic to which supersingular elliptic curve?

sage: H = hilbert_class_polynomial(-64)
sage: S = H.splitting_field(l)
sage: R = H.roots(ring = S)
sage: for r in R:
....:     j0 = r[0]
....:     E = EllipticCurve(j=j0)
....:     I = S.ideal(p)
....:     Ebar = E.reduction(I)
....:     Ebar
....:     Ebar.j_invariant()
Elliptic Curve defined by y^2 = x^3 + (67*lbar+26)*x + (54*lbar+71) over Residue field in lbar of Fractional ideal (179)
lbar
Elliptic Curve defined by y^2 = x^3 + (112*lbar+44)*x + (125*lbar+171) over Residue field in lbar of Fractional ideal (179)
178*lbar + 35

sage: p = 179
sage: F.<i> = GF(p^2, modulus=x^2+1)
sage: Rp = H.roots(ring=F)
sage: for r in Rp:
....:     j0 = r[0]
....:     EllipticCurve(j=j0)
....:     j0
Elliptic Curve defined by y^2 = x^3 + (10*i+35)*x + (155*i+121) over Finite Field in i of size 179^2
99*i + 107
Elliptic Curve defined by y^2 = x^3 + (169*i+35)*x + (24*i+121) over Finite Field in i of size 179^2
80*i + 107


### Reductions of elliptic curves over number fields

I have found two elliptic curves over a number field whose reductions are isomorphic to supersingular elliptic curves (used Deuring Lifting Theorem). How can I check which reduction is isomorphic to which supersingular elliptic curve?

sage: H = hilbert_class_polynomial(-64)
sage: S = H.splitting_field(l)
S.<l> = H.splitting_field()
sage: R = H.roots(ring = S)
sage: for r in R:
....:     j0 = r[0]
....:     E = EllipticCurve(j=j0)
....:     I = S.ideal(p)
....:     Ebar = E.reduction(I)
....:     Ebar
....:     Ebar.j_invariant()
Elliptic Curve defined by y^2 = x^3 + (67*lbar+26)*x + (54*lbar+71) over Residue field in lbar of Fractional ideal (179)
lbar
Elliptic Curve defined by y^2 = x^3 + (112*lbar+44)*x + (125*lbar+171) over Residue field in lbar of Fractional ideal (179)
178*lbar + 35

sage: p = 179
sage: F.<i> = GF(p^2, modulus=x^2+1)
sage: Rp = H.roots(ring=F)
sage: for r in Rp:
....:     j0 = r[0]
....:     EllipticCurve(j=j0)
....:     j0
Elliptic Curve defined by y^2 = x^3 + (10*i+35)*x + (155*i+121) over Finite Field in i of size 179^2
99*i + 107
Elliptic Curve defined by y^2 = x^3 + (169*i+35)*x + (24*i+121) over Finite Field in i of size 179^2
80*i + 107

 3 retagged FrédéricC 5141 ●3 ●43 ●112

### Reductions of elliptic curves over number fields

I have found two elliptic curves over a number field whose reductions are isomorphic to supersingular elliptic curves (used Deuring Lifting Theorem). How can I check which reduction is isomorphic to which supersingular elliptic curve?

sage: H = hilbert_class_polynomial(-64)
sage: S.<l> = H.splitting_field()
sage: R = H.roots(ring = S)
sage: for r in R:
....:     j0 = r[0]
....:     E = EllipticCurve(j=j0)
....:     I = S.ideal(p)
....:     Ebar = E.reduction(I)
....:     Ebar
....:     Ebar.j_invariant()
Elliptic Curve defined by y^2 = x^3 + (67*lbar+26)*x + (54*lbar+71) over Residue field in lbar of Fractional ideal (179)
lbar
Elliptic Curve defined by y^2 = x^3 + (112*lbar+44)*x + (125*lbar+171) over Residue field in lbar of Fractional ideal (179)
178*lbar + 35

sage: p = 179
sage: F.<i> = GF(p^2, modulus=x^2+1)
sage: Rp = H.roots(ring=F)
sage: for r in Rp:
....:     j0 = r[0]
....:     EllipticCurve(j=j0)
....:     j0
Elliptic Curve defined by y^2 = x^3 + (10*i+35)*x + (155*i+121) over Finite Field in i of size 179^2
99*i + 107
Elliptic Curve defined by y^2 = x^3 + (169*i+35)*x + (24*i+121) over Finite Field in i of size 179^2
80*i + 107