Show commands:
SageMath
E = EllipticCurve("km1")
E.isogeny_class()
Elliptic curves in class 221760.km
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.km1 | 221760mo1 | \([0, 0, 0, -192, 936]\) | \(28311552/2695\) | \(74511360\) | \([2]\) | \(86016\) | \(0.24841\) | \(\Gamma_0(N)\)-optimal |
221760.km2 | 221760mo2 | \([0, 0, 0, 228, 4464]\) | \(2963088/21175\) | \(-9367142400\) | \([2]\) | \(172032\) | \(0.59498\) |
Rank
sage: E.rank()
The elliptic curves in class 221760.km have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.km do not have complex multiplication.Modular form 221760.2.a.km
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.