Show commands:
SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 319725bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 319725.bc2 | 319725bc1 | \([1, -1, 1, -66380, -22341378]\) | \(-24137569/147175\) | \(-197228575908984375\) | \([2]\) | \(2654208\) | \(2.0023\) | \(\Gamma_0(N)\)-optimal |
| 319725.bc1 | 319725bc2 | \([1, -1, 1, -1665005, -824851128]\) | \(380920459249/888125\) | \(1190172440830078125\) | \([2]\) | \(5308416\) | \(2.3488\) |
Rank
sage: E.rank()
The elliptic curves in class 319725bc have rank \(0\).
Complex multiplication
The elliptic curves in class 319725bc do not have complex multiplication.Modular form 319725.2.a.bc
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
