Properties

Label 35490cy
Number of curves 8
Conductor 35490
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("35490.cz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 35490cy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
35490.cz7 35490cy1 [1, 0, 0, -84081, 9373545] [2] 221184 \(\Gamma_0(N)\)-optimal
35490.cz6 35490cy2 [1, 0, 0, -97601, 6153081] [2, 2] 442368  
35490.cz5 35490cy3 [1, 0, 0, -248856, -36335040] [2] 663552  
35490.cz8 35490cy4 [1, 0, 0, 324899, 45276581] [2] 884736  
35490.cz4 35490cy5 [1, 0, 0, -736421, -239026035] [2] 884736  
35490.cz2 35490cy6 [1, 0, 0, -3709976, -2750545344] [2, 2] 1327104  
35490.cz3 35490cy7 [1, 0, 0, -3439576, -3168421504] [2] 2654208  
35490.cz1 35490cy8 [1, 0, 0, -59358296, -176028284160] [2] 2654208  

Rank

sage: E.rank()
 

The elliptic curves in class 35490cy have rank \(1\).

Modular form 35490.2.a.cz

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} + q^{12} - q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} - 8q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.