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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 406560.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
406560.bm1 | 406560bm4 | \([0, -1, 0, -39205976, -94474901640]\) | \(7347751505995469192/72930375\) | \(66150711329472000\) | \([2]\) | \(19660800\) | \(2.8038\) | |
406560.bm2 | 406560bm2 | \([0, -1, 0, -3510976, -76086140]\) | \(5276930158229192/3050936350875\) | \(2767318964578542528000\) | \([2]\) | \(19660800\) | \(2.8038\) | \(\Gamma_0(N)\)-optimal* |
406560.bm3 | 406560bm1 | \([0, -1, 0, -2452226, -1473212640]\) | \(14383655824793536/45209390625\) | \(5125836368961000000\) | \([2, 2]\) | \(9830400\) | \(2.4573\) | \(\Gamma_0(N)\)-optimal* |
406560.bm4 | 406560bm3 | \([0, -1, 0, -1423121, -2720282079]\) | \(-43927191786304/415283203125\) | \(-3013425261000000000000\) | \([2]\) | \(19660800\) | \(2.8038\) |
Rank
sage: E.rank()
The elliptic curves in class 406560.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 406560.bm do not have complex multiplication.Modular form 406560.2.a.bm
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.