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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 90459r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 90459.f3 | 90459r1 | \([1, -1, 1, -7241, -203128]\) | \(389017/57\) | \(6151335295617\) | \([2]\) | \(147840\) | \(1.1789\) | \(\Gamma_0(N)\)-optimal |
| 90459.f2 | 90459r2 | \([1, -1, 1, -31046, 1910756]\) | \(30664297/3249\) | \(350626111850169\) | \([2, 2]\) | \(295680\) | \(1.5255\) | |
| 90459.f4 | 90459r3 | \([1, -1, 1, 40369, 9366482]\) | \(67419143/390963\) | \(-42192008792637003\) | \([2]\) | \(591360\) | \(1.8720\) | |
| 90459.f1 | 90459r4 | \([1, -1, 1, -483341, 129457946]\) | \(115714886617/1539\) | \(166086052981659\) | \([2]\) | \(591360\) | \(1.8720\) |
Rank
sage: E.rank()
The elliptic curves in class 90459r have rank \(0\).
Complex multiplication
The elliptic curves in class 90459r do not have complex multiplication.Modular form 90459.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
