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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 10320s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10320.a3 | 10320s1 | \([0, -1, 0, -1096, 14320]\) | \(35578826569/51600\) | \(211353600\) | \([2]\) | \(4608\) | \(0.49949\) | \(\Gamma_0(N)\)-optimal |
| 10320.a2 | 10320s2 | \([0, -1, 0, -1416, 5616]\) | \(76711450249/41602500\) | \(170403840000\) | \([2, 2]\) | \(9216\) | \(0.84606\) | |
| 10320.a1 | 10320s3 | \([0, -1, 0, -13416, -589584]\) | \(65202655558249/512820150\) | \(2100511334400\) | \([2]\) | \(18432\) | \(1.1926\) | |
| 10320.a4 | 10320s4 | \([0, -1, 0, 5464, 38640]\) | \(4403686064471/2721093750\) | \(-11145600000000\) | \([2]\) | \(18432\) | \(1.1926\) |
Rank
sage: E.rank()
The elliptic curves in class 10320s have rank \(1\).
Complex multiplication
The elliptic curves in class 10320s do not have complex multiplication.Modular form 10320.2.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 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.
