Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 232974bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232974.bp2 | 232974bp1 | \([1, -1, 1, 9019075, -12557480929]\) | \(9522140375/13502538\) | \(-115051048563540259425402\) | \([]\) | \(21845376\) | \(3.1106\) | \(\Gamma_0(N)\)-optimal |
| 232974.bp1 | 232974bp2 | \([1, -1, 1, -87581930, 521027830289]\) | \(-8719509765625/8716379112\) | \(-74269634087527835287457448\) | \([3]\) | \(65536128\) | \(3.6599\) |
Rank
sage: E.rank()
The elliptic curves in class 232974bp have rank \(0\).
Complex multiplication
The elliptic curves in class 232974bp do not have complex multiplication.Modular form 232974.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
