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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 194040ej
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 194040.u4 | 194040ej1 | \([0, 0, 0, 160377, 23682778]\) | \(20777545136/23059575\) | \(-506298956712595200\) | \([2]\) | \(1572864\) | \(2.0837\) | \(\Gamma_0(N)\)-optimal |
| 194040.u3 | 194040ej2 | \([0, 0, 0, -906843, 222399142]\) | \(939083699236/300155625\) | \(26361020060242560000\) | \([2, 2]\) | \(3145728\) | \(2.4302\) | |
| 194040.u1 | 194040ej3 | \([0, 0, 0, -13131363, 18312243838]\) | \(1425631925916578/270703125\) | \(47548737482400000000\) | \([2]\) | \(6291456\) | \(2.7768\) | |
| 194040.u2 | 194040ej4 | \([0, 0, 0, -5757843, -5149598258]\) | \(120186986927618/4332064275\) | \(760922826320747059200\) | \([2]\) | \(6291456\) | \(2.7768\) |
Rank
sage: E.rank()
The elliptic curves in class 194040ej have rank \(1\).
Complex multiplication
The elliptic curves in class 194040ej do not have complex multiplication.Modular form 194040.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
