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SageMath
E = EllipticCurve("gh1")
E.isogeny_class()
Elliptic curves in class 212160.gh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.gh1 | 212160eh4 | \([0, 1, 0, -89505, -9970497]\) | \(302503589987689/12214946250\) | \(3202074869760000\) | \([2]\) | \(1572864\) | \(1.7413\) | |
| 212160.gh2 | 212160eh2 | \([0, 1, 0, -14625, 467775]\) | \(1319778683209/395612100\) | \(103707338342400\) | \([2, 2]\) | \(786432\) | \(1.3947\) | |
| 212160.gh3 | 212160eh1 | \([0, 1, 0, -13345, 588863]\) | \(1002702430729/159120\) | \(41712353280\) | \([2]\) | \(393216\) | \(1.0481\) | \(\Gamma_0(N)\)-optimal |
| 212160.gh4 | 212160eh3 | \([0, 1, 0, 39775, 3176895]\) | \(26546265663191/31856082570\) | \(-8350880909230080\) | \([4]\) | \(1572864\) | \(1.7413\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.gh have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.gh do not have complex multiplication.Modular form 212160.2.a.gh
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 LMFDB 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 LMFDB labels.
