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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 86640ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86640.eg4 | 86640ef1 | \([0, 1, 0, 72080, -6468652]\) | \(214921799/218880\) | \(-42178160366714880\) | \([2]\) | \(1105920\) | \(1.8769\) | \(\Gamma_0(N)\)-optimal |
| 86640.eg3 | 86640ef2 | \([0, 1, 0, -390000, -59885100]\) | \(34043726521/11696400\) | \(2253895444596326400\) | \([2, 2]\) | \(2211840\) | \(2.2235\) | |
| 86640.eg2 | 86640ef3 | \([0, 1, 0, -2584880, 1554668628]\) | \(9912050027641/311647500\) | \(60054450990888960000\) | \([4]\) | \(4423680\) | \(2.5701\) | |
| 86640.eg1 | 86640ef4 | \([0, 1, 0, -5588400, -5085698220]\) | \(100162392144121/23457780\) | \(4520312530551521280\) | \([2]\) | \(4423680\) | \(2.5701\) |
Rank
sage: E.rank()
The elliptic curves in class 86640ef have rank \(1\).
Complex multiplication
The elliptic curves in class 86640ef do not have complex multiplication.Modular form 86640.2.a.ef
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.
