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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 12870.ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.ce1 | 12870cf4 | \([1, -1, 1, -343337, 77519211]\) | \(6139836723518159689/3799803150\) | \(2770056496350\) | \([2]\) | \(98304\) | \(1.7084\) | |
| 12870.ce2 | 12870cf3 | \([1, -1, 1, -48317, -2338941]\) | \(17111482619973769/6627044531250\) | \(4831115463281250\) | \([2]\) | \(98304\) | \(1.7084\) | |
| 12870.ce3 | 12870cf2 | \([1, -1, 1, -21587, 1200111]\) | \(1525998818291689/37268302500\) | \(27168592522500\) | \([2, 2]\) | \(49152\) | \(1.3618\) | |
| 12870.ce4 | 12870cf1 | \([1, -1, 1, 193, 58839]\) | \(1095912791/2055596400\) | \(-1498529775600\) | \([4]\) | \(24576\) | \(1.0153\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 12870.ce have rank \(0\).
Complex multiplication
The elliptic curves in class 12870.ce do not have complex multiplication.Modular form 12870.2.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 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.
