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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 258570cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.cf1 | 258570cf1 | \([1, -1, 0, -5304519, 4702115965]\) | \(2135227170133/832320\) | \(6434399613320677440\) | \([2]\) | \(6709248\) | \(2.5740\) | \(\Gamma_0(N)\)-optimal |
| 258570.cf2 | 258570cf2 | \([1, -1, 0, -4513599, 6152188693]\) | \(-1315451937493/1353040200\) | \(-10459920871404426263400\) | \([2]\) | \(13418496\) | \(2.9206\) |
Rank
sage: E.rank()
The elliptic curves in class 258570cf have rank \(1\).
Complex multiplication
The elliptic curves in class 258570cf do not have complex multiplication.Modular form 258570.2.a.cf
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
