Show commands:
SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 44352.ei
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44352.ei1 | 44352bz2 | \([0, 0, 0, -820524, -200651600]\) | \(1278763167594532/375974556419\) | \(17962464157987700736\) | \([2]\) | \(737280\) | \(2.4004\) | |
44352.ei2 | 44352bz1 | \([0, 0, 0, 137796, -20870768]\) | \(24226243449392/29774625727\) | \(-355626224107241472\) | \([2]\) | \(368640\) | \(2.0538\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 44352.ei have rank \(1\).
Complex multiplication
The elliptic curves in class 44352.ei do not have complex multiplication.Modular form 44352.2.a.ei
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.