Properties

Label 86190d
Number of curves 8
Conductor 86190
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("86190.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 86190d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86190.d6 86190d1 [1, 1, 0, -13523, 738093] [2] 294912 \(\Gamma_0(N)\)-optimal
86190.d5 86190d2 [1, 1, 0, -229843, 42314797] [2, 2] 589824  
86190.d4 86190d3 [1, 1, 0, -243363, 37039293] [2, 2] 1179648  
86190.d2 86190d4 [1, 1, 0, -3677443, 2712825757] [2] 1179648  
86190.d7 86190d5 [1, 1, 0, 517137, 223057593] [2] 2359296  
86190.d3 86190d6 [1, 1, 0, -1220183, -486340863] [2, 2] 2359296  
86190.d8 86190d7 [1, 1, 0, 1106947, -2119520697] [2] 4718592  
86190.d1 86190d8 [1, 1, 0, -19176433, -32329954613] [2] 4718592  

Rank

sage: E.rank()
 

The elliptic curves in class 86190d have rank \(2\).

Modular form 86190.2.a.d

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

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.