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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 40768.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40768.ec1 | 40768db2 | \([0, 0, 0, -666988, -230040496]\) | \(-1064019559329/125497034\) | \(-3870451447382933504\) | \([]\) | \(1016064\) | \(2.3025\) | |
| 40768.ec2 | 40768db1 | \([0, 0, 0, -8428, 455504]\) | \(-2146689/1664\) | \(-51319389814784\) | \([]\) | \(145152\) | \(1.3295\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40768.ec have rank \(1\).
Complex multiplication
The elliptic curves in class 40768.ec do not have complex multiplication.Modular form 40768.2.a.ec
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
