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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 86190ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86190.ch3 | 86190ce1 | \([1, 1, 1, -35240, 2531225]\) | \(1002702430729/159120\) | \(768041848080\) | \([4]\) | \(344064\) | \(1.2909\) | \(\Gamma_0(N)\)-optimal |
| 86190.ch2 | 86190ce2 | \([1, 1, 1, -38620, 2012057]\) | \(1319778683209/395612100\) | \(1909544044788900\) | \([2, 2]\) | \(688128\) | \(1.6375\) | |
| 86190.ch4 | 86190ce3 | \([1, 1, 1, 105030, 13618977]\) | \(26546265663191/31856082570\) | \(-153763226053619130\) | \([2]\) | \(1376256\) | \(1.9841\) | |
| 86190.ch1 | 86190ce4 | \([1, 1, 1, -236350, -42754015]\) | \(302503589987689/12214946250\) | \(58959212494016250\) | \([2]\) | \(1376256\) | \(1.9841\) |
Rank
sage: E.rank()
The elliptic curves in class 86190ce have rank \(0\).
Complex multiplication
The elliptic curves in class 86190ce do not have complex multiplication.Modular form 86190.2.a.ce
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.
