Show commands for:
SageMath
sage: E = EllipticCurve("i1")
sage: E.isogeny_class()
Elliptic curves in class 3465i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 3465.n6 | 3465i1 | [1, -1, 0, 315, 624456] | [2] | 7680 | \(\Gamma_0(N)\)-optimal |
| 3465.n5 | 3465i2 | [1, -1, 0, -107730, 13395375] | [2, 2] | 15360 | |
| 3465.n4 | 3465i3 | [1, -1, 0, -229005, -22186710] | [2] | 30720 | |
| 3465.n2 | 3465i4 | [1, -1, 0, -1715175, 865019736] | [2, 2] | 30720 | |
| 3465.n1 | 3465i5 | [1, -1, 0, -27442800, 55340692911] | [2] | 61440 | |
| 3465.n3 | 3465i6 | [1, -1, 0, -1706670, 874016325] | [2] | 61440 |
Rank
sage: E.rank()
The elliptic curves in class 3465i have rank \(0\).
Complex multiplication
The elliptic curves in class 3465i do not have complex multiplication.Modular form 3465.2.a.i
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
