Properties

Label 61710o
Number of curves 6
Conductor 61710
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("61710.m1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 61710o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
61710.m4 61710o1 [1, 1, 0, -884512, -320555264] [2] 737280 \(\Gamma_0(N)\)-optimal
61710.m3 61710o2 [1, 1, 0, -894192, -313192656] [2, 2] 1474560  
61710.m5 61710o3 [1, 1, 0, 463428, -1178539644] [2] 2949120  
61710.m2 61710o4 [1, 1, 0, -2406692, 1023554844] [2, 2] 2949120  
61710.m6 61710o5 [1, 1, 0, 6335558, 6760219294] [2] 5898240  
61710.m1 61710o6 [1, 1, 0, -35348942, 80868980394] [2] 5898240  

Rank

sage: E.rank()
 

The elliptic curves in class 61710o have rank \(1\).

Modular form 61710.2.a.m

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} - 6q^{13} - q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.