Properties

Label 327600.ks
Number of curves 8
Conductor 327600
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("327600.ks1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 327600.ks

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
327600.ks1 327600ks8 [0, 0, 0, -144685638075, -21182940650969750] [2] 509607936  
327600.ks2 327600ks6 [0, 0, 0, -9042858075, -330983009909750] [2, 2] 254803968  
327600.ks3 327600ks7 [0, 0, 0, -8931726075, -339514502417750] [2] 509607936  
327600.ks4 327600ks5 [0, 0, 0, -1787058075, -29029667309750] [2] 169869312  
327600.ks5 327600ks3 [0, 0, 0, -572130075, -5037867197750] [2] 127401984  
327600.ks6 327600ks2 [0, 0, 0, -149058075, -123881309750] [2, 2] 84934656  
327600.ks7 327600ks1 [0, 0, 0, -92610075, 341193762250] [2] 42467328 \(\Gamma_0(N)\)-optimal
327600.ks8 327600ks4 [0, 0, 0, 585773925, -982899917750] [2] 169869312  

Rank

sage: E.rank()
 

The elliptic curves in class 327600.ks have rank \(0\).

Modular form 327600.2.a.ks

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

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.