Show commands for:
SageMath
sage: E = EllipticCurve("ec1")
sage: E.isogeny_class()
Elliptic curves in class 248430.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 248430.ec1 | 248430ec5 | [1, 0, 1, -139120973, 631580989106] | [2] | 28311552 | |
| 248430.ec2 | 248430ec3 | [1, 0, 1, -8695223, 9867524006] | [2, 2] | 14155776 | |
| 248430.ec3 | 248430ec6 | [1, 0, 1, -8115553, 11239950698] | [2] | 28311552 | |
| 248430.ec4 | 248430ec4 | [1, 0, 1, -3064143, -1951516682] | [2] | 14155776 | |
| 248430.ec5 | 248430ec2 | [1, 0, 1, -579843, 132314158] | [2, 2] | 7077888 | |
| 248430.ec6 | 248430ec1 | [1, 0, 1, 82637, 12802766] | [2] | 3538944 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 248430.ec have rank \(2\).
Complex multiplication
The elliptic curves in class 248430.ec do not have complex multiplication.Modular form 248430.2.a.ec
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
