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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 57498.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57498.f1 | 57498b4 | \([1, 1, 0, -1839964, 959876932]\) | \(268498407453697/252\) | \(646563055068\) | \([2]\) | \(774144\) | \(1.9946\) | |
| 57498.f2 | 57498b6 | \([1, 1, 0, -1251294, -534099078]\) | \(84448510979617/933897762\) | \(2396126151269396658\) | \([2]\) | \(1548288\) | \(2.3411\) | |
| 57498.f3 | 57498b3 | \([1, 1, 0, -142404, 7261020]\) | \(124475734657/63011844\) | \(161671152230588196\) | \([2, 2]\) | \(774144\) | \(1.9946\) | |
| 57498.f4 | 57498b2 | \([1, 1, 0, -115024, 14954800]\) | \(65597103937/63504\) | \(162933889877136\) | \([2, 2]\) | \(387072\) | \(1.6480\) | |
| 57498.f5 | 57498b1 | \([1, 1, 0, -5504, 344832]\) | \(-7189057/16128\) | \(-41380035524352\) | \([2]\) | \(193536\) | \(1.3014\) | \(\Gamma_0(N)\)-optimal |
| 57498.f6 | 57498b5 | \([1, 1, 0, 528406, 56766798]\) | \(6359387729183/4218578658\) | \(-10823718671274379122\) | \([2]\) | \(1548288\) | \(2.3411\) |
Rank
sage: E.rank()
The elliptic curves in class 57498.f have rank \(1\).
Complex multiplication
The elliptic curves in class 57498.f do not have complex multiplication.Modular form 57498.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
