Show commands:
SageMath
E = EllipticCurve("kx1")
E.isogeny_class()
Elliptic curves in class 463680.kx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.kx1 | 463680kx1 | \([0, 0, 0, -94781172, -355165870264]\) | \(-126142795384287538429696/9315359375\) | \(-6953878512000000\) | \([]\) | \(34007040\) | \(2.9361\) | \(\Gamma_0(N)\)-optimal* |
463680.kx2 | 463680kx2 | \([0, 0, 0, -93827172, -362665491064]\) | \(-122372013839654770813696/5297595236711512175\) | \(-3954633653824196992588800\) | \([]\) | \(102021120\) | \(3.4854\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.kx have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.kx do not have complex multiplication.Modular form 463680.2.a.kx
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.