Show commands:
SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 55770.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55770.cd1 | 55770cj4 | \([1, 1, 1, -6447100, -6303468865]\) | \(6139836723518159689/3799803150\) | \(18340924042648350\) | \([2]\) | \(2064384\) | \(2.4416\) | |
| 55770.cd2 | 55770cj3 | \([1, 1, 1, -907280, 190925327]\) | \(17111482619973769/6627044531250\) | \(31987478186838281250\) | \([2]\) | \(2064384\) | \(2.4416\) | |
| 55770.cd3 | 55770cj2 | \([1, 1, 1, -405350, -97383265]\) | \(1525998818291689/37268302500\) | \(179886977921722500\) | \([2, 2]\) | \(1032192\) | \(2.0950\) | |
| 55770.cd4 | 55770cj1 | \([1, 1, 1, 3630, -4790193]\) | \(1095912791/2055596400\) | \(-9921971203887600\) | \([4]\) | \(516096\) | \(1.7484\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55770.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 55770.cd do not have complex multiplication.Modular form 55770.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
