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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 3648.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3648.bc1 | 3648bh3 | \([0, 1, 0, -27393, 1735935]\) | \(8671983378625/82308\) | \(21576548352\) | \([2]\) | \(6912\) | \(1.1452\) | |
3648.bc2 | 3648bh4 | \([0, 1, 0, -26753, 1821567]\) | \(-8078253774625/846825858\) | \(-221990317719552\) | \([2]\) | \(13824\) | \(1.4918\) | |
3648.bc3 | 3648bh1 | \([0, 1, 0, -513, -513]\) | \(57066625/32832\) | \(8606711808\) | \([2]\) | \(2304\) | \(0.59592\) | \(\Gamma_0(N)\)-optimal |
3648.bc4 | 3648bh2 | \([0, 1, 0, 2047, -2049]\) | \(3616805375/2105352\) | \(-551905394688\) | \([2]\) | \(4608\) | \(0.94249\) |
Rank
sage: E.rank()
The elliptic curves in class 3648.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 3648.bc do not have complex multiplication.Modular form 3648.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.