Show commands:
SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 55770l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55770.s3 | 55770l1 | \([1, 1, 0, -11157, -332691]\) | \(31824875809/8785920\) | \(42407957729280\) | \([2]\) | \(193536\) | \(1.3223\) | \(\Gamma_0(N)\)-optimal |
55770.s2 | 55770l2 | \([1, 1, 0, -65237, 6124461]\) | \(6361447449889/294465600\) | \(1421329208270400\) | \([2, 2]\) | \(387072\) | \(1.6689\) | |
55770.s4 | 55770l3 | \([1, 1, 0, 36163, 23544981]\) | \(1083523132511/50179392120\) | \(-242206341499345080\) | \([2]\) | \(774144\) | \(2.0155\) | |
55770.s1 | 55770l4 | \([1, 1, 0, -1031917, 403043269]\) | \(25176685646263969/57915000\) | \(279544643235000\) | \([2]\) | \(774144\) | \(2.0155\) |
Rank
sage: E.rank()
The elliptic curves in class 55770l have rank \(0\).
Complex multiplication
The elliptic curves in class 55770l do not have complex multiplication.Modular form 55770.2.a.l
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.