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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 258570.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.ec1 | 258570ec3 | \([1, -1, 1, -9948893, 12080748957]\) | \(30949975477232209/478125000\) | \(1682399360728125000\) | \([2]\) | \(14155776\) | \(2.6323\) | |
| 258570.ec2 | 258570ec2 | \([1, -1, 1, -640373, 177013581]\) | \(8253429989329/936360000\) | \(3294810908049960000\) | \([2, 2]\) | \(7077888\) | \(2.2857\) | |
| 258570.ec3 | 258570ec1 | \([1, -1, 1, -153653, -20205363]\) | \(114013572049/15667200\) | \(55128862252339200\) | \([2]\) | \(3538944\) | \(1.9392\) | \(\Gamma_0(N)\)-optimal |
| 258570.ec4 | 258570ec4 | \([1, -1, 1, 880627, 889449981]\) | \(21464092074671/109596256200\) | \(-385641142732707568200\) | \([2]\) | \(14155776\) | \(2.6323\) |
Rank
sage: E.rank()
The elliptic curves in class 258570.ec have rank \(1\).
Complex multiplication
The elliptic curves in class 258570.ec do not have complex multiplication.Modular form 258570.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
