Properties

Label 36414.cg
Number of curves 6
Conductor 36414
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("36414.cg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 36414.cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36414.cg1 36414cf6 [1, -1, 1, -7102085, 7286730621] [2] 663552  
36414.cg2 36414cf5 [1, -1, 1, -443525, 114129789] [2] 331776  
36414.cg3 36414cf4 [1, -1, 1, -92390, 8882925] [2] 221184  
36414.cg4 36414cf2 [1, -1, 1, -27365, -1734357] [2] 73728  
36414.cg5 36414cf1 [1, -1, 1, -1355, -38505] [2] 36864 \(\Gamma_0(N)\)-optimal
36414.cg6 36414cf3 [1, -1, 1, 11650, 809421] [2] 110592  

Rank

sage: E.rank()
 

The elliptic curves in class 36414.cg have rank \(1\).

Modular form 36414.2.a.cg

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{7} + q^{8} - 4q^{13} - q^{14} + q^{16} + 2q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.