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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 49098.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
49098.bp1 | 49098bo4 | \([1, 0, 0, -10697142, 13465133100]\) | \(1150638118585800835537/31752757008504\) | \(3735680109293487096\) | \([2]\) | \(3096576\) | \(2.6668\) | |
49098.bp2 | 49098bo3 | \([1, 0, 0, -2986502, -1796616228]\) | \(25039399590518087377/2641281025170312\) | \(310744071330262036488\) | \([2]\) | \(3096576\) | \(2.6668\) | |
49098.bp3 | 49098bo2 | \([1, 0, 0, -695262, 192638340]\) | \(315922815546536017/46479778841664\) | \(5468299500942927936\) | \([2, 2]\) | \(1548288\) | \(2.3203\) | |
49098.bp4 | 49098bo1 | \([1, 0, 0, 73058, 16385732]\) | \(366554400441263/1197281046528\) | \(-140858917842972672\) | \([2]\) | \(774144\) | \(1.9737\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 49098.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 49098.bp do not have complex multiplication.Modular form 49098.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.