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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 14520.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14520.bm1 | 14520v5 | \([0, 1, 0, -387240, 92622000]\) | \(1770025017602/75\) | \(272111769600\) | \([2]\) | \(81920\) | \(1.6789\) | |
14520.bm2 | 14520v3 | \([0, 1, 0, -24240, 1436400]\) | \(868327204/5625\) | \(10204191360000\) | \([2, 2]\) | \(40960\) | \(1.3324\) | |
14520.bm3 | 14520v6 | \([0, 1, 0, -9720, 3155568]\) | \(-27995042/1171875\) | \(-4251746400000000\) | \([2]\) | \(81920\) | \(1.6789\) | |
14520.bm4 | 14520v2 | \([0, 1, 0, -2460, -9792]\) | \(3631696/2025\) | \(918377222400\) | \([2, 2]\) | \(20480\) | \(0.98580\) | |
14520.bm5 | 14520v1 | \([0, 1, 0, -1855, -31330]\) | \(24918016/45\) | \(1275523920\) | \([2]\) | \(10240\) | \(0.63923\) | \(\Gamma_0(N)\)-optimal |
14520.bm6 | 14520v4 | \([0, 1, 0, 9640, -67872]\) | \(54607676/32805\) | \(-59510844011520\) | \([2]\) | \(40960\) | \(1.3324\) |
Rank
sage: E.rank()
The elliptic curves in class 14520.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 14520.bm do not have complex multiplication.Modular form 14520.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.