Properties

Label 51842.j
Number of curves 6
Conductor 51842
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("51842.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51842.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51842.j1 51842e6 [1, 1, 0, -70777830, -229218601324] [2] 3649536  
51842.j2 51842e5 [1, 1, 0, -4420070, -3588945772] [2] 1824768  
51842.j3 51842e4 [1, 1, 0, -920735, -279119203] [2] 1216512  
51842.j4 51842e2 [1, 1, 0, -272710, 54665514] [2] 405504  
51842.j5 51842e1 [1, 1, 0, -13500, 1216412] [2] 202752 \(\Gamma_0(N)\)-optimal
51842.j6 51842e3 [1, 1, 0, 116105, -25508139] [2] 608256  

Rank

sage: E.rank()
 

The elliptic curves in class 51842.j have rank \(1\).

Modular form 51842.2.a.j

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.