Show commands for:
SageMath
sage: E = EllipticCurve("40425.p1")
sage: E.isogeny_class()
Elliptic curves in class 40425y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
40425.p3 | 40425y1 | [1, 1, 1, -7988, 269156] | [2] | 55296 | \(\Gamma_0(N)\)-optimal |
40425.p2 | 40425y2 | [1, 1, 1, -14113, -208594] | [2, 2] | 110592 | |
40425.p4 | 40425y3 | [1, 1, 1, 53262, -1556094] | [2] | 221184 | |
40425.p1 | 40425y4 | [1, 1, 1, -179488, -29314594] | [2] | 221184 |
Rank
sage: E.rank()
The elliptic curves in class 40425y have rank \(1\).
Modular form 40425.2.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 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.