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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 63175n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63175.u3 | 63175n1 | \([1, -1, 0, -128042, 17634991]\) | \(315821241/665\) | \(488836107265625\) | \([2]\) | \(276480\) | \(1.7036\) | \(\Gamma_0(N)\)-optimal |
| 63175.u2 | 63175n2 | \([1, -1, 0, -173167, 4142616]\) | \(781229961/442225\) | \(325076011331640625\) | \([2, 2]\) | \(552960\) | \(2.0502\) | |
| 63175.u4 | 63175n3 | \([1, -1, 0, 684208, 32435991]\) | \(48188806119/28511875\) | \(-20958848099013671875\) | \([2]\) | \(1105920\) | \(2.3968\) | |
| 63175.u1 | 63175n4 | \([1, -1, 0, -1752542, -888204259]\) | \(809818183161/4561235\) | \(3352926859734921875\) | \([2]\) | \(1105920\) | \(2.3968\) |
Rank
sage: E.rank()
The elliptic curves in class 63175n have rank \(1\).
Complex multiplication
The elliptic curves in class 63175n do not have complex multiplication.Modular form 63175.2.a.n
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.
