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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 6240.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6240.p1 | 6240z2 | \([0, -1, 0, -227520, -41682600]\) | \(2543984126301795848/909361981125\) | \(465593334336000\) | \([2]\) | \(49152\) | \(1.7838\) | |
| 6240.p2 | 6240z3 | \([0, -1, 0, -117520, 15226900]\) | \(350584567631475848/8259273550125\) | \(4228748057664000\) | \([4]\) | \(49152\) | \(1.7838\) | |
| 6240.p3 | 6240z1 | \([0, -1, 0, -16270, -446600]\) | \(7442744143086784/2927948765625\) | \(187388721000000\) | \([2, 2]\) | \(24576\) | \(1.4372\) | \(\Gamma_0(N)\)-optimal |
| 6240.p4 | 6240z4 | \([0, -1, 0, 52175, -3280223]\) | \(3834800837445824/3342041015625\) | \(-13689000000000000\) | \([4]\) | \(49152\) | \(1.7838\) |
Rank
sage: E.rank()
The elliptic curves in class 6240.p have rank \(0\).
Complex multiplication
The elliptic curves in class 6240.p do not have complex multiplication.Modular form 6240.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
