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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 480240bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
480240.bp4 | 480240bp1 | \([0, 0, 0, 6117, 342538]\) | \(8477185319/21880935\) | \(-65336121815040\) | \([2]\) | \(983040\) | \(1.3346\) | \(\Gamma_0(N)\)-optimal* |
480240.bp3 | 480240bp2 | \([0, 0, 0, -52203, 3853402]\) | \(5268932332201/900900225\) | \(2690073657446400\) | \([2, 2]\) | \(1966080\) | \(1.6812\) | \(\Gamma_0(N)\)-optimal* |
480240.bp1 | 480240bp3 | \([0, 0, 0, -797403, 274062922]\) | \(18778886261717401/732035835\) | \(2185847290736640\) | \([2]\) | \(3932160\) | \(2.0278\) | \(\Gamma_0(N)\)-optimal* |
480240.bp2 | 480240bp4 | \([0, 0, 0, -240123, -41660822]\) | \(512787603508921/45649063125\) | \(136307372106240000\) | \([2]\) | \(3932160\) | \(2.0278\) |
Rank
sage: E.rank()
The elliptic curves in class 480240bp have rank \(1\).
Complex multiplication
The elliptic curves in class 480240bp do not have complex multiplication.Modular form 480240.2.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.