Show commands:
SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 450800.ga
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450800.ga1 | 450800ga3 | \([0, -1, 0, -33144008, -72798473488]\) | \(534774372149809/5323062500\) | \(40080190724000000000000\) | \([2]\) | \(47775744\) | \(3.1557\) | |
450800.ga2 | 450800ga4 | \([0, -1, 0, -8644008, -178050473488]\) | \(-9486391169809/1813439640250\) | \(-13654359055089424000000000\) | \([2]\) | \(95551488\) | \(3.5023\) | |
450800.ga3 | 450800ga1 | \([0, -1, 0, -2960008, 1904182512]\) | \(380920459249/12622400\) | \(95040815206400000000\) | \([2]\) | \(15925248\) | \(2.6064\) | \(\Gamma_0(N)\)-optimal* |
450800.ga4 | 450800ga2 | \([0, -1, 0, 959992, 6576822512]\) | \(12994449551/2489452840\) | \(-18744424779082240000000\) | \([2]\) | \(31850496\) | \(2.9530\) |
Rank
sage: E.rank()
The elliptic curves in class 450800.ga have rank \(1\).
Complex multiplication
The elliptic curves in class 450800.ga do not have complex multiplication.Modular form 450800.2.a.ga
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.