Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 259182.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259182.bp1 | 259182bp2 | \([1, -1, 0, -4362, 115352]\) | \(-3854428875/137564\) | \(-327628337652\) | \([]\) | \(352512\) | \(0.98169\) | |
259182.bp2 | 259182bp1 | \([1, -1, 0, 258, 468]\) | \(580078125/373184\) | \(-1219192128\) | \([]\) | \(117504\) | \(0.43238\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 259182.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 259182.bp do not have complex multiplication.Modular form 259182.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.