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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 20691q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20691.p3 | 20691q1 | \([1, -1, 0, -1656, -22005]\) | \(389017/57\) | \(73613674233\) | \([2]\) | \(17280\) | \(0.81010\) | \(\Gamma_0(N)\)-optimal |
| 20691.p2 | 20691q2 | \([1, -1, 0, -7101, 209952]\) | \(30664297/3249\) | \(4195979431281\) | \([2, 2]\) | \(34560\) | \(1.1567\) | |
| 20691.p1 | 20691q3 | \([1, -1, 0, -110556, 14176377]\) | \(115714886617/1539\) | \(1987569204291\) | \([2]\) | \(69120\) | \(1.5032\) | |
| 20691.p4 | 20691q4 | \([1, -1, 0, 9234, 1023435]\) | \(67419143/390963\) | \(-504916191564147\) | \([2]\) | \(69120\) | \(1.5032\) |
Rank
sage: E.rank()
The elliptic curves in class 20691q have rank \(0\).
Complex multiplication
The elliptic curves in class 20691q do not have complex multiplication.Modular form 20691.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
