Show commands:
SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 364560.ga
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 364560.ga1 | 364560ga4 | \([0, 1, 0, -498209280, -4280383341900]\) | \(28379906689597370652529/1357352437500\) | \(654094978742016000000\) | \([2]\) | \(74649600\) | \(3.4714\) | |
| 364560.ga2 | 364560ga3 | \([0, 1, 0, -31086400, -67121813452]\) | \(-6894246873502147249/47925198774000\) | \(-23094688606463287296000\) | \([2]\) | \(37324800\) | \(3.1248\) | |
| 364560.ga3 | 364560ga2 | \([0, 1, 0, -6688320, -4786537932]\) | \(68663623745397169/19216056254400\) | \(9260031190113917337600\) | \([2]\) | \(24883200\) | \(2.9221\) | |
| 364560.ga4 | 364560ga1 | \([0, 1, 0, 1088960, -490368460]\) | \(296354077829711/387386634240\) | \(-186677862939450408960\) | \([2]\) | \(12441600\) | \(2.5755\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 364560.ga have rank \(0\).
Complex multiplication
The elliptic curves in class 364560.ga do not have complex multiplication.Modular form 364560.2.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.
