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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 283920.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 283920.e1 | 283920e5 | \([0, -1, 0, -18414296, 30415148880]\) | \(69855246474511682/14613770535\) | \(144461576483374725120\) | \([2]\) | \(16515072\) | \(2.8645\) | |
| 283920.e2 | 283920e4 | \([0, -1, 0, -8206696, -9046256480]\) | \(12367124507424964/14926275\) | \(73775389190630400\) | \([2]\) | \(8257536\) | \(2.5179\) | |
| 283920.e3 | 283920e3 | \([0, -1, 0, -1277696, 364407120]\) | \(46670944188964/15429366225\) | \(76261994249345049600\) | \([2, 2]\) | \(8257536\) | \(2.5179\) | |
| 283920.e4 | 283920e2 | \([0, -1, 0, -517196, -138739680]\) | \(12381975627856/419225625\) | \(518021637063840000\) | \([2, 2]\) | \(4128768\) | \(2.1713\) | |
| 283920.e5 | 283920e1 | \([0, -1, 0, 10929, -7553430]\) | \(1869154304/319921875\) | \(-24707228568750000\) | \([2]\) | \(2064384\) | \(1.8248\) | \(\Gamma_0(N)\)-optimal |
| 283920.e6 | 283920e6 | \([0, -1, 0, 3690904, 2502892560]\) | \(562511980386718/599562079935\) | \(-5926853925865425745920\) | \([2]\) | \(16515072\) | \(2.8645\) |
Rank
sage: E.rank()
The elliptic curves in class 283920.e have rank \(1\).
Complex multiplication
The elliptic curves in class 283920.e do not have complex multiplication.Modular form 283920.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
