Show commands:
SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 84672.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
84672.bk1 | 84672ks3 | \([0, 0, 0, -89964, -10520496]\) | \(-132651/2\) | \(-1214085997264896\) | \([]\) | \(435456\) | \(1.6975\) | |
84672.bk2 | 84672ks2 | \([0, 0, 0, -42924, 4538576]\) | \(-1167051/512\) | \(-3837111299997696\) | \([]\) | \(435456\) | \(1.6975\) | |
84672.bk3 | 84672ks1 | \([0, 0, 0, 4116, -71344]\) | \(9261/8\) | \(-6661651562496\) | \([]\) | \(145152\) | \(1.1482\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84672.bk have rank \(0\).
Complex multiplication
The elliptic curves in class 84672.bk do not have complex multiplication.Modular form 84672.2.a.bk
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.