Show commands:
SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 10304.s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10304.s1 | 10304h3 | \([0, 0, 0, -7916, 271024]\) | \(209267191953/55223\) | \(14476378112\) | \([2]\) | \(10240\) | \(0.93409\) | |
10304.s2 | 10304h2 | \([0, 0, 0, -556, 3120]\) | \(72511713/25921\) | \(6795034624\) | \([2, 2]\) | \(5120\) | \(0.58751\) | |
10304.s3 | 10304h1 | \([0, 0, 0, -236, -1360]\) | \(5545233/161\) | \(42205184\) | \([2]\) | \(2560\) | \(0.24094\) | \(\Gamma_0(N)\)-optimal |
10304.s4 | 10304h4 | \([0, 0, 0, 1684, 21936]\) | \(2014698447/1958887\) | \(-513510473728\) | \([2]\) | \(10240\) | \(0.93409\) |
Rank
sage: E.rank()
The elliptic curves in class 10304.s have rank \(2\).
Complex multiplication
The elliptic curves in class 10304.s do not have complex multiplication.Modular form 10304.2.a.s
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.