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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 8112.g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8112.g1 | 8112y4 | \([0, -1, 0, -1135944, -465618960]\) | \(18013780041269221/9216\) | \(82933972992\) | \([2]\) | \(57600\) | \(1.8632\) | |
| 8112.g2 | 8112y3 | \([0, -1, 0, -70984, -7260176]\) | \(-4395631034341/3145728\) | \(-28308129447936\) | \([2]\) | \(28800\) | \(1.5166\) | |
| 8112.g3 | 8112y2 | \([0, -1, 0, -3384, 29808]\) | \(476379541/236196\) | \(2125507018752\) | \([2]\) | \(11520\) | \(1.0584\) | |
| 8112.g4 | 8112y1 | \([0, -1, 0, 776, 3184]\) | \(5735339/3888\) | \(-34987769856\) | \([2]\) | \(5760\) | \(0.71186\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8112.g have rank \(1\).
Complex multiplication
The elliptic curves in class 8112.g do not have complex multiplication.Modular form 8112.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
