Properties

Label 436800.hy
Number of curves 8
Conductor 436800
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("436800.hy1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 436800.hy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
436800.hy1 436800hy7 [0, -1, 0, -64304728033, 6276448294455937] [u'2'] 509607936 \(\Gamma_0(N)\)-optimal*
436800.hy2 436800hy6 [0, -1, 0, -4019048033, 98070379655937] [u'2', u'2'] 254803968 \(\Gamma_0(N)\)-optimal*
436800.hy3 436800hy8 [0, -1, 0, -3969656033, 100598212823937] [u'2'] 509607936  
436800.hy4 436800hy4 [0, -1, 0, -794248033, 8601647655937] [u'2'] 169869312 \(\Gamma_0(N)\)-optimal*
436800.hy5 436800hy3 [0, -1, 0, -254280033, 1492786151937] [u'2'] 127401984 \(\Gamma_0(N)\)-optimal*
436800.hy6 436800hy2 [0, -1, 0, -66248033, 36727655937] [u'2', u'2'] 84934656 \(\Gamma_0(N)\)-optimal*
436800.hy7 436800hy1 [0, -1, 0, -41160033, -101080728063] [u'2'] 42467328 \(\Gamma_0(N)\)-optimal*
436800.hy8 436800hy5 [0, -1, 0, 260343967, 291142823937] [u'2'] 169869312  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 436800.hy7.

Rank

sage: E.rank()
 

The elliptic curves in class 436800.hy have rank \(2\).

Modular form 436800.2.a.hy

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.