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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 289578a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 289578.a2 | 289578a1 | \([1, 1, 0, -1306, -76076]\) | \(-10218313/96192\) | \(-2321841037248\) | \([2]\) | \(709632\) | \(1.0552\) | \(\Gamma_0(N)\)-optimal |
| 289578.a1 | 289578a2 | \([1, 1, 0, -35986, -2635460]\) | \(213525509833/669336\) | \(16156143884184\) | \([2]\) | \(1419264\) | \(1.4018\) |
Rank
sage: E.rank()
The elliptic curves in class 289578a have rank \(1\).
Complex multiplication
The elliptic curves in class 289578a do not have complex multiplication.Modular form 289578.2.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.
