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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 20691r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20691.a2 | 20691r1 | \([0, 0, 1, 21417, -1377918]\) | \(841232384/1121931\) | \(-1448937949928139\) | \([]\) | \(129600\) | \(1.5949\) | \(\Gamma_0(N)\)-optimal |
| 20691.a1 | 20691r2 | \([0, 0, 1, -4781073, -4023806328]\) | \(-9358714467168256/22284891\) | \(-28780222919156379\) | \([]\) | \(648000\) | \(2.3997\) |
Rank
sage: E.rank()
The elliptic curves in class 20691r have rank \(0\).
Complex multiplication
The elliptic curves in class 20691r do not have complex multiplication.Modular form 20691.2.a.r
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
