Properties

Label 436800mr
Number of curves 8
Conductor 436800
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("436800.mr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 436800mr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
436800.mr7 436800mr1 [0, 1, 0, -41160033, 101080728063] [u'2'] 42467328 \(\Gamma_0(N)\)-optimal
436800.mr6 436800mr2 [0, 1, 0, -66248033, -36727655937] [u'2', u'2'] 84934656  
436800.mr5 436800mr3 [0, 1, 0, -254280033, -1492786151937] [u'2'] 127401984  
436800.mr8 436800mr4 [0, 1, 0, 260343967, -291142823937] [u'2'] 169869312  
436800.mr4 436800mr5 [0, 1, 0, -794248033, -8601647655937] [u'2'] 169869312  
436800.mr2 436800mr6 [0, 1, 0, -4019048033, -98070379655937] [u'2', u'2'] 254803968  
436800.mr3 436800mr7 [0, 1, 0, -3969656033, -100598212823937] [u'2'] 509607936  
436800.mr1 436800mr8 [0, 1, 0, -64304728033, -6276448294455937] [u'2'] 509607936  

Rank

sage: E.rank()
 

The elliptic curves in class 436800mr have rank \(1\).

Modular form 436800.2.a.mr

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 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.