Show commands:
SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 86640ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86640.o2 | 86640ca1 | \([0, -1, 0, -453536, -412203264]\) | \(-53540005609/350208000\) | \(-67485056586743808000\) | \([2]\) | \(2903040\) | \(2.4883\) | \(\Gamma_0(N)\)-optimal |
86640.o1 | 86640ca2 | \([0, -1, 0, -11543456, -15059769600]\) | \(882774443450089/2166000000\) | \(417388045295616000000\) | \([2]\) | \(5806080\) | \(2.8349\) |
Rank
sage: E.rank()
The elliptic curves in class 86640ca have rank \(1\).
Complex multiplication
The elliptic curves in class 86640ca do not have complex multiplication.Modular form 86640.2.a.ca
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.