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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2695a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2695.c4 | 2695a1 | \([1, -1, 0, 40, -85]\) | \(59319/55\) | \(-6470695\) | \([2]\) | \(384\) | \(-0.0058576\) | \(\Gamma_0(N)\)-optimal |
| 2695.c3 | 2695a2 | \([1, -1, 0, -205, -624]\) | \(8120601/3025\) | \(355888225\) | \([2, 2]\) | \(768\) | \(0.34072\) | |
| 2695.c1 | 2695a3 | \([1, -1, 0, -2900, -59375]\) | \(22930509321/6875\) | \(808836875\) | \([2]\) | \(1536\) | \(0.68729\) | |
| 2695.c2 | 2695a4 | \([1, -1, 0, -1430, 20691]\) | \(2749884201/73205\) | \(8612495045\) | \([2]\) | \(1536\) | \(0.68729\) |
Rank
sage: E.rank()
The elliptic curves in class 2695a have rank \(0\).
Complex multiplication
The elliptic curves in class 2695a do not have complex multiplication.Modular form 2695.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
