Properties

Label 27735g
Number of curves 8
Conductor 27735
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("27735.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 27735g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27735.k7 27735g1 [1, 0, 1, -39, -14819] [2] 20160 \(\Gamma_0(N)\)-optimal
27735.k6 27735g2 [1, 0, 1, -9284, -340243] [2, 2] 40320  
27735.k5 27735g3 [1, 0, 1, -18529, 447431] [2, 2] 80640  
27735.k4 27735g4 [1, 0, 1, -147959, -21918073] [2] 80640  
27735.k8 27735g5 [1, 0, 1, 64676, 3376247] [2] 161280  
27735.k2 27735g6 [1, 0, 1, -249654, 47966731] [2, 2] 161280  
27735.k3 27735g7 [1, 0, 1, -203429, 66290321] [2] 322560  
27735.k1 27735g8 [1, 0, 1, -3993879, 3071802841] [2] 322560  

Rank

sage: E.rank()
 

The elliptic curves in class 27735g have rank \(1\).

Modular form 27735.2.a.k

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3q^{8} + q^{9} - q^{10} - 4q^{11} - q^{12} - 2q^{13} - q^{15} - q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.