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SageMath
E = EllipticCurve("gl1")
E.isogeny_class()
Elliptic curves in class 212160gl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.cx3 | 212160gl1 | \([0, -1, 0, -1905, 28017]\) | \(46689225424/7249905\) | \(118782443520\) | \([2]\) | \(196608\) | \(0.84829\) | \(\Gamma_0(N)\)-optimal |
| 212160.cx2 | 212160gl2 | \([0, -1, 0, -8385, -266175]\) | \(994958062276/98903025\) | \(6481708646400\) | \([2, 2]\) | \(393216\) | \(1.1949\) | |
| 212160.cx4 | 212160gl3 | \([0, -1, 0, 10335, -1303263]\) | \(931329171502/6107473125\) | \(-800518717440000\) | \([4]\) | \(786432\) | \(1.5414\) | |
| 212160.cx1 | 212160gl4 | \([0, -1, 0, -130785, -18161055]\) | \(1887517194957938/21849165\) | \(2863813754880\) | \([2]\) | \(786432\) | \(1.5414\) |
Rank
sage: E.rank()
The elliptic curves in class 212160gl have rank \(2\).
Complex multiplication
The elliptic curves in class 212160gl do not have complex multiplication.Modular form 212160.2.a.gl
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.
