Show commands:
SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 85680.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85680.bf1 | 85680dp4 | \([0, 0, 0, -146163, 16597618]\) | \(115650783909361/27072079335\) | \(80836795741040640\) | \([2]\) | \(786432\) | \(1.9561\) | |
85680.bf2 | 85680dp2 | \([0, 0, 0, -48963, -3950462]\) | \(4347507044161/258084225\) | \(770635366502400\) | \([2, 2]\) | \(393216\) | \(1.6095\) | |
85680.bf3 | 85680dp1 | \([0, 0, 0, -48243, -4078478]\) | \(4158523459441/16065\) | \(47969832960\) | \([2]\) | \(196608\) | \(1.2630\) | \(\Gamma_0(N)\)-optimal |
85680.bf4 | 85680dp3 | \([0, 0, 0, 36717, -16305518]\) | \(1833318007919/39525924375\) | \(-118023777768960000\) | \([2]\) | \(786432\) | \(1.9561\) |
Rank
sage: E.rank()
The elliptic curves in class 85680.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 85680.bf do not have complex multiplication.Modular form 85680.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.