Properties

Label 86190bm
Number of curves 8
Conductor 86190
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("86190.bm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 86190bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
86190.bm7 86190bm1 [1, 0, 1, -354248678, 2471124782648] [2] 46448640 \(\Gamma_0(N)\)-optimal
86190.bm6 86190bm2 [1, 0, 1, -939177958, -7792278335944] [2, 2] 92897280  
86190.bm5 86190bm3 [1, 0, 1, -4358980838, -110049775142344] [2] 139345920  
86190.bm8 86190bm4 [1, 0, 1, 2514857562, -51791164403912] [4] 185794560  
86190.bm4 86190bm5 [1, 0, 1, -13752081958, -620648852140744] [2] 185794560  
86190.bm2 86190bm6 [1, 0, 1, -69617438518, -7070099217161992] [2, 2] 278691840  
86190.bm3 86190bm7 [1, 0, 1, -69491198898, -7097017240838744] [4] 557383680  
86190.bm1 86190bm8 [1, 0, 1, -1113879001018, -452486084448911992] [2] 557383680  

Rank

sage: E.rank()
 

The elliptic curves in class 86190bm have rank \(0\).

Modular form 86190.2.a.bm

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.