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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 56350.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56350.bp1 | 56350bf1 | \([1, -1, 1, -1630, 13997]\) | \(89314623/36800\) | \(197225000000\) | \([2]\) | \(55296\) | \(0.86471\) | \(\Gamma_0(N)\)-optimal |
56350.bp2 | 56350bf2 | \([1, -1, 1, 5370, 97997]\) | \(3196010817/2645000\) | \(-14175546875000\) | \([2]\) | \(110592\) | \(1.2113\) |
Rank
sage: E.rank()
The elliptic curves in class 56350.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 56350.bp do not have complex multiplication.Modular form 56350.2.a.bp
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.