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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 59290.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
59290.bc1 | 59290br4 | \([1, -1, 0, -1587119, -769160925]\) | \(2121328796049/120050\) | \(25021106729684450\) | \([2]\) | \(983040\) | \(2.2109\) | |
59290.bc2 | 59290br3 | \([1, -1, 0, -519899, 134964143]\) | \(74565301329/5468750\) | \(1139809891111718750\) | \([2]\) | \(983040\) | \(2.2109\) | |
59290.bc3 | 59290br2 | \([1, -1, 0, -104869, -10545375]\) | \(611960049/122500\) | \(25531741560902500\) | \([2, 2]\) | \(491520\) | \(1.8643\) | |
59290.bc4 | 59290br1 | \([1, -1, 0, 13711, -987827]\) | \(1367631/2800\) | \(-583582664249200\) | \([2]\) | \(245760\) | \(1.5177\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 59290.bc have rank \(2\).
Complex multiplication
The elliptic curves in class 59290.bc do not have complex multiplication.Modular form 59290.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.