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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 61710co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61710.cj4 | 61710co1 | \([1, 0, 0, -2266151, -1313237595]\) | \(726497538898787209/1038579300\) | \(1839906583287300\) | \([2]\) | \(1382400\) | \(2.2003\) | \(\Gamma_0(N)\)-optimal |
| 61710.cj3 | 61710co2 | \([1, 0, 0, -2286721, -1288187449]\) | \(746461053445307689/27443694341250\) | \(48618178590879191250\) | \([2]\) | \(2764800\) | \(2.5469\) | |
| 61710.cj2 | 61710co3 | \([1, 0, 0, -2885066, -539845404]\) | \(1499114720492202169/796539777000000\) | \(1411118803881897000000\) | \([2]\) | \(4147200\) | \(2.7496\) | |
| 61710.cj1 | 61710co4 | \([1, 0, 0, -26663986, 52587017660]\) | \(1183430669265454849849/10449720703125000\) | \(18512317658548828125000\) | \([2]\) | \(8294400\) | \(3.0962\) |
Rank
sage: E.rank()
The elliptic curves in class 61710co have rank \(1\).
Complex multiplication
The elliptic curves in class 61710co do not have complex multiplication.Modular form 61710.2.a.co
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
