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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 52800gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52800.dy4 | 52800gp1 | \([0, 1, 0, -3233, -2337]\) | \(912673/528\) | \(2162688000000\) | \([2]\) | \(98304\) | \(1.0563\) | \(\Gamma_0(N)\)-optimal |
| 52800.dy2 | 52800gp2 | \([0, 1, 0, -35233, 2525663]\) | \(1180932193/4356\) | \(17842176000000\) | \([2, 2]\) | \(196608\) | \(1.4029\) | |
| 52800.dy3 | 52800gp3 | \([0, 1, 0, -19233, 4845663]\) | \(-192100033/2371842\) | \(-9715064832000000\) | \([2]\) | \(393216\) | \(1.7495\) | |
| 52800.dy1 | 52800gp4 | \([0, 1, 0, -563233, 162509663]\) | \(4824238966273/66\) | \(270336000000\) | \([2]\) | \(393216\) | \(1.7495\) |
Rank
sage: E.rank()
The elliptic curves in class 52800gp have rank \(1\).
Complex multiplication
The elliptic curves in class 52800gp do not have complex multiplication.Modular form 52800.2.a.gp
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.
