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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 3570k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3570.k4 | 3570k1 | \([1, 0, 1, 536, -538]\) | \(17075848639751/10028415600\) | \(-10028415600\) | \([6]\) | \(2304\) | \(0.60845\) | \(\Gamma_0(N)\)-optimal |
| 3570.k3 | 3570k2 | \([1, 0, 1, -2164, -4858]\) | \(1119971462469049/638680075740\) | \(638680075740\) | \([6]\) | \(4608\) | \(0.95502\) | |
| 3570.k2 | 3570k3 | \([1, 0, 1, -7879, -283894]\) | \(-54082626581000809/3358656000000\) | \(-3358656000000\) | \([2]\) | \(6912\) | \(1.1578\) | |
| 3570.k1 | 3570k4 | \([1, 0, 1, -127879, -17611894]\) | \(231268521845235080809/816013464000\) | \(816013464000\) | \([2]\) | \(13824\) | \(1.5043\) |
Rank
sage: E.rank()
The elliptic curves in class 3570k have rank \(0\).
Complex multiplication
The elliptic curves in class 3570k do not have complex multiplication.Modular form 3570.2.a.k
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.
