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SageMath
E = EllipticCurve("cl1")
E.isogeny_class()
Elliptic curves in class 78078.cl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78078.cl1 | 78078bw4 | \([1, 1, 1, -38282, -2897731]\) | \(1285429208617/614922\) | \(2968111043898\) | \([2]\) | \(294912\) | \(1.3481\) | |
| 78078.cl2 | 78078bw3 | \([1, 1, 1, -21382, 1174493]\) | \(223980311017/4278582\) | \(20651898104838\) | \([2]\) | \(294912\) | \(1.3481\) | |
| 78078.cl3 | 78078bw2 | \([1, 1, 1, -2792, -30139]\) | \(498677257/213444\) | \(1030253420196\) | \([2, 2]\) | \(147456\) | \(1.0016\) | |
| 78078.cl4 | 78078bw1 | \([1, 1, 1, 588, -3099]\) | \(4657463/3696\) | \(-17839886064\) | \([2]\) | \(73728\) | \(0.65499\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78078.cl have rank \(0\).
Complex multiplication
The elliptic curves in class 78078.cl do not have complex multiplication.Modular form 78078.2.a.cl
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
