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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 39600f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39600.cl5 | 39600f1 | \([0, 0, 0, -39450, 5040875]\) | \(-37256083456/38671875\) | \(-7047949218750000\) | \([2]\) | \(196608\) | \(1.7363\) | \(\Gamma_0(N)\)-optimal |
| 39600.cl4 | 39600f2 | \([0, 0, 0, -742575, 246212750]\) | \(15529488955216/6125625\) | \(17862322500000000\) | \([2, 2]\) | \(393216\) | \(2.0828\) | |
| 39600.cl3 | 39600f3 | \([0, 0, 0, -855075, 166675250]\) | \(5927735656804/2401490025\) | \(28010979651600000000\) | \([2, 2]\) | \(786432\) | \(2.4294\) | |
| 39600.cl1 | 39600f4 | \([0, 0, 0, -11880075, 15760750250]\) | \(15897679904620804/2475\) | \(28868400000000\) | \([2]\) | \(786432\) | \(2.4294\) | |
| 39600.cl6 | 39600f5 | \([0, 0, 0, 2789925, 1212790250]\) | \(102949393183198/86815346805\) | \(-2025228410267040000000\) | \([2]\) | \(1572864\) | \(2.7760\) | |
| 39600.cl2 | 39600f6 | \([0, 0, 0, -6300075, -5969839750]\) | \(1185450336504002/26043266205\) | \(607537314030240000000\) | \([2]\) | \(1572864\) | \(2.7760\) |
Rank
sage: E.rank()
The elliptic curves in class 39600f have rank \(0\).
Complex multiplication
The elliptic curves in class 39600f do not have complex multiplication.Modular form 39600.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
