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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 443982bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
443982.bp4 | 443982bp1 | \([1, 1, 1, -93237, -40110261]\) | \(-100999381393/723148272\) | \(-641796753308789232\) | \([2]\) | \(5806080\) | \(2.1001\) | \(\Gamma_0(N)\)-optimal* |
443982.bp3 | 443982bp2 | \([1, 1, 1, -2418857, -1445714989]\) | \(1763535241378513/4612311396\) | \(4093443341868248676\) | \([2, 2]\) | \(11612160\) | \(2.4467\) | \(\Gamma_0(N)\)-optimal* |
443982.bp2 | 443982bp3 | \([1, 1, 1, -3370247, -203960761]\) | \(4770223741048753/2740574865798\) | \(2432270281451806002438\) | \([2]\) | \(23224320\) | \(2.7932\) | \(\Gamma_0(N)\)-optimal* |
443982.bp1 | 443982bp4 | \([1, 1, 1, -38677387, -92599659409]\) | \(7209828390823479793/49509306\) | \(43939691318755386\) | \([2]\) | \(23224320\) | \(2.7932\) |
Rank
sage: E.rank()
The elliptic curves in class 443982bp have rank \(1\).
Complex multiplication
The elliptic curves in class 443982bp do not have complex multiplication.Modular form 443982.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.