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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 162624cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162624.u3 | 162624cn1 | \([0, -1, 0, -10204, 400138]\) | \(1036433728/63\) | \(7142933952\) | \([2]\) | \(184320\) | \(0.95132\) | \(\Gamma_0(N)\)-optimal |
| 162624.u2 | 162624cn2 | \([0, -1, 0, -10809, 350649]\) | \(19248832/3969\) | \(28800309694464\) | \([2, 2]\) | \(368640\) | \(1.2979\) | |
| 162624.u4 | 162624cn3 | \([0, -1, 0, 23071, 2078529]\) | \(23393656/45927\) | \(-2666085811716096\) | \([2]\) | \(737280\) | \(1.6445\) | |
| 162624.u1 | 162624cn4 | \([0, -1, 0, -54369, -4554207]\) | \(306182024/21609\) | \(1254413488914432\) | \([2]\) | \(737280\) | \(1.6445\) |
Rank
sage: E.rank()
The elliptic curves in class 162624cn have rank \(1\).
Complex multiplication
The elliptic curves in class 162624cn do not have complex multiplication.Modular form 162624.2.a.cn
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.
