Show commands:
SageMath
E = EllipticCurve("hk1")
E.isogeny_class()
Elliptic curves in class 47040hk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.hk2 | 47040hk1 | \([0, 1, 0, -16480, -874630]\) | \(-65743598656/5294205\) | \(-39862907138880\) | \([2]\) | \(184320\) | \(1.3561\) | \(\Gamma_0(N)\)-optimal |
47040.hk1 | 47040hk2 | \([0, 1, 0, -268585, -53665417]\) | \(4446542056384/25725\) | \(12396628070400\) | \([2]\) | \(368640\) | \(1.7027\) |
Rank
sage: E.rank()
The elliptic curves in class 47040hk have rank \(1\).
Complex multiplication
The elliptic curves in class 47040hk do not have complex multiplication.Modular form 47040.2.a.hk
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.