Properties

Label 31680ee
Number of curves 4
Conductor 31680
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("31680.eh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 31680ee

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
31680.eh4 31680ee1 [0, 0, 0, -1632, 24856] [2] 27648 \(\Gamma_0(N)\)-optimal
31680.eh3 31680ee2 [0, 0, 0, -3612, -47216] [2] 55296  
31680.eh2 31680ee3 [0, 0, 0, -16032, -771464] [2] 82944  
31680.eh1 31680ee4 [0, 0, 0, -255612, -49741616] [2] 165888  

Rank

sage: E.rank()
 

The elliptic curves in class 31680ee have rank \(1\).

Modular form 31680.2.a.eh

sage: E.q_eigenform(10)
 
\( q + q^{5} + 4q^{7} + q^{11} + 4q^{13} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.