Properties

Label 286650mt
Number of curves 8
Conductor 286650
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("286650.mt1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 286650mt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
286650.mt7 286650mt1 [1, -1, 1, -283618355, 1828656224147] [2] 84934656 \(\Gamma_0(N)\)-optimal
286650.mt6 286650mt2 [1, -1, 1, -456490355, -663812271853] [2, 2] 169869312  
286650.mt5 286650mt3 [1, -1, 1, -1752148355, -26999381475853] [2] 254803968  
286650.mt8 286650mt4 [1, -1, 1, 1793932645, -5268177729853] [2] 339738624  
286650.mt4 286650mt5 [1, -1, 1, -5472865355, -155579505021853] [2] 339738624  
286650.mt2 286650mt6 [1, -1, 1, -27693752855, -1773855145296853] [2, 2] 509607936  
286650.mt3 286650mt7 [1, -1, 1, -27353411105, -1819578698042353] [2] 1019215872  
286650.mt1 286650mt8 [1, -1, 1, -443099766605, -113527211776349353] [2] 1019215872  

Rank

sage: E.rank()
 

The elliptic curves in class 286650mt have rank \(1\).

Modular form 286650.2.a.mt

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

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.