Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 216384.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 216384.r1 | 216384cq3 | \([0, -1, 0, -27264449, 54804375585]\) | \(581400938887066376/13041\) | \(50274651635712\) | \([2]\) | \(7077888\) | \(2.6053\) | |
| 216384.r2 | 216384cq4 | \([0, -1, 0, -1862849, 687663873]\) | \(185446537613576/54423757521\) | \(209810248452935811072\) | \([2]\) | \(7077888\) | \(2.6053\) | |
| 216384.r3 | 216384cq2 | \([0, -1, 0, -1704089, 856679769]\) | \(1135671162482368/170067681\) | \(81953966497665024\) | \([2, 2]\) | \(3538944\) | \(2.2587\) | |
| 216384.r4 | 216384cq1 | \([0, -1, 0, -96644, 15986034]\) | \(-13258203533632/6930522081\) | \(-52183615507684416\) | \([2]\) | \(1769472\) | \(1.9121\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 216384.r have rank \(2\).
Complex multiplication
The elliptic curves in class 216384.r do not have complex multiplication.Modular form 216384.2.a.r
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.
