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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 236992.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
236992.ca1 | 236992ca2 | \([0, -1, 0, -11934945, -15866079679]\) | \(38758598383688/25921\) | \(125738623918702592\) | \([2]\) | \(15138816\) | \(2.5974\) | |
236992.ca2 | 236992ca1 | \([0, -1, 0, -741305, -250951879]\) | \(-74299881664/1958887\) | \(-1187780929517744128\) | \([2]\) | \(7569408\) | \(2.2508\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 236992.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 236992.ca do not have complex multiplication.Modular form 236992.2.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 LMFDB 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 LMFDB labels.