Properties

Label 38640.j
Number of curves $4$
Conductor $38640$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 38640.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.j1 38640b4 \([0, -1, 0, -220816, -39865184]\) \(581416486276209698/12425175\) \(25446758400\) \([2]\) \(131072\) \(1.5244\)  
38640.j2 38640b2 \([0, -1, 0, -13816, -617984]\) \(284840777767396/1312250625\) \(1343744640000\) \([2, 2]\) \(65536\) \(1.1778\)  
38640.j3 38640b3 \([0, -1, 0, -6816, -1250784]\) \(-17101973157698/321306440175\) \(-658035589478400\) \([2]\) \(131072\) \(1.5244\)  
38640.j4 38640b1 \([0, -1, 0, -1316, 2016]\) \(985329269584/566015625\) \(144900000000\) \([2]\) \(32768\) \(0.83124\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 38640.j have rank \(0\).

Complex multiplication

The elliptic curves in class 38640.j do not have complex multiplication.

Modular form 38640.2.a.j

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} + 2q^{13} + q^{15} + 6q^{17} - 4q^{19} + O(q^{20})\)  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.