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SageMath
E = EllipticCurve("hx1")
E.isogeny_class()
Elliptic curves in class 277200.hx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277200.hx1 | 277200hx4 | \([0, 0, 0, -11159175, 7565103250]\) | \(52702650535889104/22020583921875\) | \(64212022716187500000000\) | \([2]\) | \(23887872\) | \(3.0734\) | |
| 277200.hx2 | 277200hx2 | \([0, 0, 0, -9620175, 11484774250]\) | \(33766427105425744/9823275\) | \(28644669900000000\) | \([2]\) | \(7962624\) | \(2.5241\) | |
| 277200.hx3 | 277200hx1 | \([0, 0, 0, -598800, 180991375]\) | \(-130287139815424/2250652635\) | \(-410181442728750000\) | \([2]\) | \(3981312\) | \(2.1776\) | \(\Gamma_0(N)\)-optimal |
| 277200.hx4 | 277200hx3 | \([0, 0, 0, 2317200, 867344875]\) | \(7549996227362816/6152409907875\) | \(-1121276705710218750000\) | \([2]\) | \(11943936\) | \(2.7269\) |
Rank
sage: E.rank()
The elliptic curves in class 277200.hx have rank \(0\).
Complex multiplication
The elliptic curves in class 277200.hx do not have complex multiplication.Modular form 277200.2.a.hx
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
