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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 34656.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34656.be1 | 34656r4 | \([0, 1, 0, -11672, 481320]\) | \(7301384/3\) | \(72262473216\) | \([2]\) | \(55296\) | \(1.0460\) | |
34656.be2 | 34656r3 | \([0, 1, 0, -6257, -189057]\) | \(140608/3\) | \(578099785728\) | \([2]\) | \(55296\) | \(1.0460\) | |
34656.be3 | 34656r1 | \([0, 1, 0, -842, 4800]\) | \(21952/9\) | \(27098427456\) | \([2, 2]\) | \(27648\) | \(0.69943\) | \(\Gamma_0(N)\)-optimal |
34656.be4 | 34656r2 | \([0, 1, 0, 2768, 38012]\) | \(97336/81\) | \(-1951086776832\) | \([2]\) | \(55296\) | \(1.0460\) |
Rank
sage: E.rank()
The elliptic curves in class 34656.be have rank \(1\).
Complex multiplication
The elliptic curves in class 34656.be do not have complex multiplication.Modular form 34656.2.a.be
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.