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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 69300bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69300.z3 | 69300bj1 | \([0, 0, 0, -598800, -180991375]\) | \(-130287139815424/2250652635\) | \(-410181442728750000\) | \([2]\) | \(995328\) | \(2.1776\) | \(\Gamma_0(N)\)-optimal |
| 69300.z2 | 69300bj2 | \([0, 0, 0, -9620175, -11484774250]\) | \(33766427105425744/9823275\) | \(28644669900000000\) | \([2]\) | \(1990656\) | \(2.5241\) | |
| 69300.z4 | 69300bj3 | \([0, 0, 0, 2317200, -867344875]\) | \(7549996227362816/6152409907875\) | \(-1121276705710218750000\) | \([2]\) | \(2985984\) | \(2.7269\) | |
| 69300.z1 | 69300bj4 | \([0, 0, 0, -11159175, -7565103250]\) | \(52702650535889104/22020583921875\) | \(64212022716187500000000\) | \([2]\) | \(5971968\) | \(3.0734\) |
Rank
sage: E.rank()
The elliptic curves in class 69300bj have rank \(0\).
Complex multiplication
The elliptic curves in class 69300bj do not have complex multiplication.Modular form 69300.2.a.bj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
