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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 69360.ci
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69360.ci1 | 69360x6 | \([0, 1, 0, -924896, 342055380]\) | \(1770025017602/75\) | \(3707530598400\) | \([2]\) | \(589824\) | \(1.8966\) | |
| 69360.ci2 | 69360x4 | \([0, 1, 0, -57896, 5312580]\) | \(868327204/5625\) | \(139032397440000\) | \([2, 2]\) | \(294912\) | \(1.5500\) | |
| 69360.ci3 | 69360x5 | \([0, 1, 0, -23216, 11652084]\) | \(-27995042/1171875\) | \(-57930165600000000\) | \([2]\) | \(589824\) | \(1.8966\) | |
| 69360.ci4 | 69360x2 | \([0, 1, 0, -5876, -35076]\) | \(3631696/2025\) | \(12512915769600\) | \([2, 2]\) | \(147456\) | \(1.2035\) | |
| 69360.ci5 | 69360x1 | \([0, 1, 0, -4431, -114840]\) | \(24918016/45\) | \(17379049680\) | \([2]\) | \(73728\) | \(0.85688\) | \(\Gamma_0(N)\)-optimal |
| 69360.ci6 | 69360x3 | \([0, 1, 0, 23024, -254716]\) | \(54607676/32805\) | \(-810836941870080\) | \([2]\) | \(294912\) | \(1.5500\) |
Rank
sage: E.rank()
The elliptic curves in class 69360.ci have rank \(0\).
Complex multiplication
The elliptic curves in class 69360.ci do not have complex multiplication.Modular form 69360.2.a.ci
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
