Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 75456.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75456.c1 | 75456dk2 | \([0, 0, 0, -83244, -7847984]\) | \(333822098953/53954184\) | \(10310805130051584\) | \([]\) | \(700416\) | \(1.7948\) | |
75456.c2 | 75456dk1 | \([0, 0, 0, -22764, 1320784]\) | \(6826561273/7074\) | \(1351862452224\) | \([]\) | \(233472\) | \(1.2455\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75456.c have rank \(2\).
Complex multiplication
The elliptic curves in class 75456.c do not have complex multiplication.Modular form 75456.2.a.c
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.