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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 119952ej
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 119952.dm2 | 119952ej1 | \([0, 0, 0, -1802955, -931459718]\) | \(1845026709625/793152\) | \(278632736372293632\) | \([2]\) | \(1658880\) | \(2.3075\) | \(\Gamma_0(N)\)-optimal |
| 119952.dm3 | 119952ej2 | \([0, 0, 0, -1520715, -1232948486]\) | \(-1107111813625/1228691592\) | \(-431636937732729372672\) | \([2]\) | \(3317760\) | \(2.6541\) | |
| 119952.dm1 | 119952ej3 | \([0, 0, 0, -5295675, 3546270826]\) | \(46753267515625/11591221248\) | \(4071973208447954976768\) | \([2]\) | \(4976640\) | \(2.8568\) | |
| 119952.dm4 | 119952ej4 | \([0, 0, 0, 12767685, 22487510122]\) | \(655215969476375/1001033261568\) | \(-351661014370982861733888\) | \([2]\) | \(9953280\) | \(3.2034\) |
Rank
sage: E.rank()
The elliptic curves in class 119952ej have rank \(0\).
Complex multiplication
The elliptic curves in class 119952ej do not have complex multiplication.Modular form 119952.2.a.ej
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
