Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 45570.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45570.a1 | 45570g2 | \([1, 1, 0, -98613, 4135293]\) | \(901456690969801/457629750000\) | \(53839682457750000\) | \([2]\) | \(691200\) | \(1.9029\) | |
45570.a2 | 45570g1 | \([1, 1, 0, 22907, 513997]\) | \(11298232190519/7472736000\) | \(-879159917664000\) | \([2]\) | \(345600\) | \(1.5563\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 45570.a have rank \(1\).
Complex multiplication
The elliptic curves in class 45570.a do not have complex multiplication.Modular form 45570.2.a.a
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.