Properties

Label 35490x
Number of curves 8
Conductor 35490
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("35490.z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 35490x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
35490.z7 35490x1 [1, 0, 1, -4347529, 3470020556] [2] 1548288 \(\Gamma_0(N)\)-optimal
35490.z6 35490x2 [1, 0, 1, -6997449, -1260616628] [2, 2] 3096576  
35490.z5 35490x3 [1, 0, 1, -26858329, -51243878164] [2] 4644864  
35490.z8 35490x4 [1, 0, 1, 27498831, -9995074724] [2] 6193152  
35490.z4 35490x5 [1, 0, 1, -83892449, -295276338628] [2] 6193152  
35490.z2 35490x6 [1, 0, 1, -424511949, -3366561638828] [2, 2] 9289728  
35490.z3 35490x7 [1, 0, 1, -419294919, -3453337542224] [2] 18579456  
35490.z1 35490x8 [1, 0, 1, -6792186899, -215458531803448] [2] 18579456  

Rank

sage: E.rank()
 

The elliptic curves in class 35490x have rank \(0\).

Modular form 35490.2.a.z

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.