Properties

Label 162624.hy
Number of curves 6
Conductor 162624
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("162624.hy1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 162624.hy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
162624.hy1 162624gh5 [0, 1, 0, -34995297, 79670741247] [2] 9830400  
162624.hy2 162624gh3 [0, 1, 0, -2199457, 1229651135] [2, 2] 4915200  
162624.hy3 162624gh2 [0, 1, 0, -302177, -35834625] [2, 2] 2457600  
162624.hy4 162624gh1 [0, 1, 0, -263457, -52120257] [2] 1228800 \(\Gamma_0(N)\)-optimal
162624.hy5 162624gh6 [0, 1, 0, 239903, 3810006143] [2] 9830400  
162624.hy6 162624gh4 [0, 1, 0, 975583, -258420417] [2] 4915200  

Rank

sage: E.rank()
 

The elliptic curves in class 162624.hy have rank \(1\).

Modular form 162624.2.a.hy

sage: E.q_eigenform(10)
 
\( q + q^{3} + 2q^{5} - q^{7} + q^{9} + 6q^{13} + 2q^{15} - 2q^{17} + 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.