Show commands:
SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 57330.du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
57330.du1 | 57330eh4 | \([1, -1, 1, -213233, -37839049]\) | \(12501706118329/2570490\) | \(220460956369290\) | \([2]\) | \(393216\) | \(1.7493\) | |
57330.du2 | 57330eh2 | \([1, -1, 1, -14783, -451069]\) | \(4165509529/1368900\) | \(117405243036900\) | \([2, 2]\) | \(196608\) | \(1.4027\) | |
57330.du3 | 57330eh1 | \([1, -1, 1, -5963, 173387]\) | \(273359449/9360\) | \(802770892560\) | \([2]\) | \(98304\) | \(1.0561\) | \(\Gamma_0(N)\)-optimal |
57330.du4 | 57330eh3 | \([1, -1, 1, 42547, -3134113]\) | \(99317171591/106616250\) | \(-9144062198066250\) | \([2]\) | \(393216\) | \(1.7493\) |
Rank
sage: E.rank()
The elliptic curves in class 57330.du have rank \(1\).
Complex multiplication
The elliptic curves in class 57330.du do not have complex multiplication.Modular form 57330.2.a.du
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.