Properties

Label 57330.du
Number of curves $4$
Conductor $57330$
CM no
Rank $1$
Graph

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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)
 
\(q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + q^{13} + q^{16} - 6 q^{17} + O(q^{20})\) Copy content Toggle raw display

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.