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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 139230.ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139230.ep1 | 139230h3 | \([1, -1, 1, -3437222, 2453643969]\) | \(6160540455434488353049/107450752500\) | \(78331598572500\) | \([2]\) | \(2883584\) | \(2.2065\) | |
| 139230.ep2 | 139230h4 | \([1, -1, 1, -322142, -3771039]\) | \(5071506329733538969/2926108608384780\) | \(2133133175512504620\) | \([2]\) | \(2883584\) | \(2.2065\) | |
| 139230.ep3 | 139230h2 | \([1, -1, 1, -215042, 38297841]\) | \(1508565467598193369/6280737699600\) | \(4578657783008400\) | \([2, 2]\) | \(1441792\) | \(1.8600\) | |
| 139230.ep4 | 139230h1 | \([1, -1, 1, -6962, 1176369]\) | \(-51184652297689/788010612480\) | \(-574459736497920\) | \([2]\) | \(720896\) | \(1.5134\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 139230.ep have rank \(0\).
Complex multiplication
The elliptic curves in class 139230.ep do not have complex multiplication.Modular form 139230.2.a.ep
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
