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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 377377j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 377377.j4 | 377377j1 | \([1, -1, 0, 98749, 13588680]\) | \(22062729659823/29354283343\) | \(-141687519028542487\) | \([2]\) | \(2580480\) | \(1.9769\) | \(\Gamma_0(N)\)-optimal |
| 377377.j3 | 377377j2 | \([1, -1, 0, -611896, 133403427]\) | \(5249244962308257/1448621666569\) | \(6992220097790248321\) | \([2, 2]\) | \(5160960\) | \(2.3234\) | |
| 377377.j1 | 377377j3 | \([1, -1, 0, -9017111, 10423067630]\) | \(16798320881842096017/2132227789307\) | \(10291856283477131363\) | \([2]\) | \(10321920\) | \(2.6700\) | |
| 377377.j2 | 377377j4 | \([1, -1, 0, -3577001, -2496644708]\) | \(1048626554636928177/48569076788309\) | \(234433656963500975981\) | \([2]\) | \(10321920\) | \(2.6700\) |
Rank
sage: E.rank()
The elliptic curves in class 377377j have rank \(0\).
Complex multiplication
The elliptic curves in class 377377j do not have complex multiplication.Modular form 377377.2.a.j
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.
