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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 7056x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7056.bo4 | 7056x1 | \([0, 0, 0, 441, -18522]\) | \(432/7\) | \(-153692888832\) | \([2]\) | \(6144\) | \(0.82708\) | \(\Gamma_0(N)\)-optimal |
| 7056.bo3 | 7056x2 | \([0, 0, 0, -8379, -277830]\) | \(740772/49\) | \(4303400887296\) | \([2, 2]\) | \(12288\) | \(1.1737\) | |
| 7056.bo1 | 7056x3 | \([0, 0, 0, -131859, -18429390]\) | \(1443468546/7\) | \(1229543110656\) | \([2]\) | \(24576\) | \(1.5202\) | |
| 7056.bo2 | 7056x4 | \([0, 0, 0, -26019, 1278018]\) | \(11090466/2401\) | \(421733286955008\) | \([2]\) | \(24576\) | \(1.5202\) |
Rank
sage: E.rank()
The elliptic curves in class 7056x have rank \(1\).
Complex multiplication
The elliptic curves in class 7056x do not have complex multiplication.Modular form 7056.2.a.x
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.
