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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 2730ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2730.z3 | 2730ba1 | \([1, 0, 0, -29946, -1997100]\) | \(2969894891179808929/22997520\) | \(22997520\) | \([2]\) | \(5120\) | \(1.0047\) | \(\Gamma_0(N)\)-optimal |
| 2730.z2 | 2730ba2 | \([1, 0, 0, -29966, -1994304]\) | \(2975849362756797409/8263842596100\) | \(8263842596100\) | \([2, 2]\) | \(10240\) | \(1.3512\) | |
| 2730.z1 | 2730ba3 | \([1, 0, 0, -42116, -227694]\) | \(8261629364934163009/4759323790524030\) | \(4759323790524030\) | \([2]\) | \(20480\) | \(1.6978\) | |
| 2730.z4 | 2730ba4 | \([1, 0, 0, -18136, -3581890]\) | \(-659704930833045889/5156082432978750\) | \(-5156082432978750\) | \([2]\) | \(20480\) | \(1.6978\) |
Rank
sage: E.rank()
The elliptic curves in class 2730ba have rank \(0\).
Complex multiplication
The elliptic curves in class 2730ba do not have complex multiplication.Modular form 2730.2.a.ba
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.
