Show commands:
SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 127296.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127296.bk1 | 127296ci4 | \([0, 0, 0, -509196, 139854224]\) | \(305612563186948/663\) | \(31675318272\) | \([2]\) | \(786432\) | \(1.6888\) | |
127296.bk2 | 127296ci3 | \([0, 0, 0, -41196, 794576]\) | \(161838334948/87947613\) | \(4201762644099072\) | \([2]\) | \(786432\) | \(1.6888\) | |
127296.bk3 | 127296ci2 | \([0, 0, 0, -31836, 2183600]\) | \(298766385232/439569\) | \(5250184003584\) | \([2, 2]\) | \(393216\) | \(1.3422\) | |
127296.bk4 | 127296ci1 | \([0, 0, 0, -1416, 54200]\) | \(-420616192/1456611\) | \(-1087354285056\) | \([2]\) | \(196608\) | \(0.99561\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 127296.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 127296.bk do not have complex multiplication.Modular form 127296.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.