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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 18207.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18207.e1 | 18207e5 | \([1, -1, 0, -2039238, 1121366389]\) | \(53297461115137/147\) | \(2586654306747\) | \([2]\) | \(147456\) | \(2.0401\) | |
| 18207.e2 | 18207e4 | \([1, -1, 0, -127503, 17530600]\) | \(13027640977/21609\) | \(380238183091809\) | \([2, 2]\) | \(73728\) | \(1.6935\) | |
| 18207.e3 | 18207e3 | \([1, -1, 0, -101493, -12344486]\) | \(6570725617/45927\) | \(808144709836527\) | \([2]\) | \(73728\) | \(1.6935\) | |
| 18207.e4 | 18207e6 | \([1, -1, 0, -88488, 28431391]\) | \(-4354703137/17294403\) | \(-304317292534477803\) | \([2]\) | \(147456\) | \(2.0401\) | |
| 18207.e5 | 18207e2 | \([1, -1, 0, -10458, 90895]\) | \(7189057/3969\) | \(69839666282169\) | \([2, 2]\) | \(36864\) | \(1.3470\) | |
| 18207.e6 | 18207e1 | \([1, -1, 0, 2547, 10264]\) | \(103823/63\) | \(-1108566131463\) | \([2]\) | \(18432\) | \(1.0004\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18207.e have rank \(1\).
Complex multiplication
The elliptic curves in class 18207.e do not have complex multiplication.Modular form 18207.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
