Show commands:
SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 268770y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
268770.y3 | 268770y1 | \([1, 0, 1, -31363, -1903234]\) | \(141339344329/17141760\) | \(413760414781440\) | \([2]\) | \(1474560\) | \(1.5352\) | \(\Gamma_0(N)\)-optimal |
268770.y2 | 268770y2 | \([1, 0, 1, -123843, 14780158]\) | \(8702409880009/1120910400\) | \(27056052122817600\) | \([2, 2]\) | \(2949120\) | \(1.8818\) | |
268770.y1 | 268770y3 | \([1, 0, 1, -1915643, 1020338318]\) | \(32208729120020809/658986840\) | \(15906340320591960\) | \([2]\) | \(5898240\) | \(2.2283\) | |
268770.y4 | 268770y4 | \([1, 0, 1, 188277, 77329006]\) | \(30579142915511/124675335000\) | \(-3009359501160615000\) | \([2]\) | \(5898240\) | \(2.2283\) |
Rank
sage: E.rank()
The elliptic curves in class 268770y have rank \(0\).
Complex multiplication
The elliptic curves in class 268770y do not have complex multiplication.Modular form 268770.2.a.y
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.