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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 17640.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17640.cp1 | 17640be5 | \([0, 0, 0, -1411347, -645355186]\) | \(1770025017602/75\) | \(13173676185600\) | \([2]\) | \(196608\) | \(2.0023\) | |
| 17640.cp2 | 17640be3 | \([0, 0, 0, -88347, -10050586]\) | \(868327204/5625\) | \(494012856960000\) | \([2, 2]\) | \(98304\) | \(1.6557\) | |
| 17640.cp3 | 17640be6 | \([0, 0, 0, -35427, -21978754]\) | \(-27995042/1171875\) | \(-205838690400000000\) | \([2]\) | \(196608\) | \(2.0023\) | |
| 17640.cp4 | 17640be2 | \([0, 0, 0, -8967, 62426]\) | \(3631696/2025\) | \(44461157126400\) | \([2, 2]\) | \(49152\) | \(1.3091\) | |
| 17640.cp5 | 17640be1 | \([0, 0, 0, -6762, 213689]\) | \(24918016/45\) | \(61751607120\) | \([2]\) | \(24576\) | \(0.96254\) | \(\Gamma_0(N)\)-optimal |
| 17640.cp6 | 17640be4 | \([0, 0, 0, 35133, 494606]\) | \(54607676/32805\) | \(-2881082981790720\) | \([2]\) | \(98304\) | \(1.6557\) |
Rank
sage: E.rank()
The elliptic curves in class 17640.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 17640.cp do not have complex multiplication.Modular form 17640.2.a.cp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
