Properties

Label 53040bp
Number of curves 8
Conductor 53040
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("53040.s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 53040bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53040.s7 53040bp1 [0, -1, 0, -33538336, -71975108864] [2] 6635520 \(\Gamma_0(N)\)-optimal
53040.s6 53040bp2 [0, -1, 0, -88916256, 227021356800] [2, 2] 13271040  
53040.s5 53040bp3 [0, -1, 0, -412684576, 3205946556160] [2] 19906560  
53040.s8 53040bp4 [0, -1, 0, 238093024, 1508636126976] [2] 26542080  
53040.s4 53040bp5 [0, -1, 0, -1301972256, 18080294342400] [4] 26542080  
53040.s2 53040bp6 [0, -1, 0, -6591000096, 205958491312896] [2, 2] 39813120  
53040.s3 53040bp7 [0, -1, 0, -6579048416, 206742626695680] [2] 79626240  
53040.s1 53040bp8 [0, -1, 0, -105456000096, 13181238367312896] [4] 79626240  

Rank

sage: E.rank()
 

The elliptic curves in class 53040bp have rank \(0\).

Modular form 53040.2.a.s

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

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.