Properties

Label 106470fe
Number of curves 8
Conductor 106470
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("106470.fd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 106470fe

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
106470.fd7 106470fe1 [1, -1, 1, -39127757, -93690555019] [2] 12386304 \(\Gamma_0(N)\)-optimal
106470.fd6 106470fe2 [1, -1, 1, -62977037, 34036648949] [2, 2] 24772608  
106470.fd5 106470fe3 [1, -1, 1, -241724957, 1383584710421] [2] 37158912  
106470.fd8 106470fe4 [1, -1, 1, 247489483, 269867017541] [2] 49545216  
106470.fd4 106470fe5 [1, -1, 1, -755032037, 7972461142949] [2] 49545216  
106470.fd2 106470fe6 [1, -1, 1, -3820607537, 90897164248349] [2, 2] 74317824  
106470.fd3 106470fe7 [1, -1, 1, -3773654267, 93240113640041] [2] 148635648  
106470.fd1 106470fe8 [1, -1, 1, -61129682087, 5817380358693089] [2] 148635648  

Rank

sage: E.rank()
 

The elliptic curves in class 106470fe have rank \(0\).

Modular form 106470.2.a.fd

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

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.