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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 13680.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13680.c1 | 13680k3 | \([0, 0, 0, -295563, -61847638]\) | \(3825131988299044/961875\) | \(718035840000\) | \([2]\) | \(81920\) | \(1.6501\) | |
| 13680.c2 | 13680k2 | \([0, 0, 0, -18543, -958642]\) | \(3778298043856/59213025\) | \(11050571577600\) | \([2, 2]\) | \(40960\) | \(1.3036\) | |
| 13680.c3 | 13680k1 | \([0, 0, 0, -2298, 19307]\) | \(115060504576/52780005\) | \(615625978320\) | \([2]\) | \(20480\) | \(0.95700\) | \(\Gamma_0(N)\)-optimal |
| 13680.c4 | 13680k4 | \([0, 0, 0, -1443, -2658382]\) | \(-445138564/4089438495\) | \(-3052749478763520\) | \([2]\) | \(81920\) | \(1.6501\) |
Rank
sage: E.rank()
The elliptic curves in class 13680.c have rank \(0\).
Complex multiplication
The elliptic curves in class 13680.c do not have complex multiplication.Modular form 13680.2.a.c
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.
