Properties

Label 40656.ce
Number of curves 6
Conductor 40656
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("40656.ce1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 40656.ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
40656.ce1 40656dk6 [0, 1, 0, -8748824, -9963217068] [2] 1228800  
40656.ce2 40656dk4 [0, 1, 0, -549864, -153981324] [2, 2] 614400  
40656.ce3 40656dk2 [0, 1, 0, -75544, 4441556] [2, 2] 307200  
40656.ce4 40656dk1 [0, 1, 0, -65864, 6482100] [2] 153600 \(\Gamma_0(N)\)-optimal
40656.ce5 40656dk5 [0, 1, 0, 59976, -476220780] [2] 1228800  
40656.ce6 40656dk3 [0, 1, 0, 243896, 32424500] [2] 614400  

Rank

sage: E.rank()
 

The elliptic curves in class 40656.ce have rank \(1\).

Modular form 40656.2.a.ce

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.