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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 40656.bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40656.bj1 | 40656bj4 | \([0, -1, 0, -77918232, 264758160240]\) | \(7209828390823479793/49509306\) | \(359255063128743936\) | \([2]\) | \(2211840\) | \(2.9683\) | |
40656.bj2 | 40656bj3 | \([0, -1, 0, -6789592, 581450608]\) | \(4770223741048753/2740574865798\) | \(19886471372095367897088\) | \([2]\) | \(2211840\) | \(2.9683\) | |
40656.bj3 | 40656bj2 | \([0, -1, 0, -4872952, 4132601200]\) | \(1763535241378513/4612311396\) | \(33468379090981502976\) | \([2, 2]\) | \(1105920\) | \(2.6217\) | |
40656.bj4 | 40656bj1 | \([0, -1, 0, -187832, 114642288]\) | \(-100999381393/723148272\) | \(-5247390826056056832\) | \([2]\) | \(552960\) | \(2.2752\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40656.bj have rank \(1\).
Complex multiplication
The elliptic curves in class 40656.bj do not have complex multiplication.Modular form 40656.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.