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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 12502a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12502.a1 | 12502a1 | \([1, -1, 0, -323, 2305]\) | \(3733252610697/23278724\) | \(23278724\) | \([2]\) | \(3360\) | \(0.25140\) | \(\Gamma_0(N)\)-optimal |
| 12502.a2 | 12502a2 | \([1, -1, 0, -133, 4851]\) | \(-261284780457/9875692358\) | \(-9875692358\) | \([2]\) | \(6720\) | \(0.59797\) |
Rank
sage: E.rank()
The elliptic curves in class 12502a have rank \(1\).
Complex multiplication
The elliptic curves in class 12502a do not have complex multiplication.Modular form 12502.2.a.a
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
