Properties

Label 68544.bq
Number of curves $6$
Conductor $68544$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("68544.bq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 68544.bq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
68544.bq1 68544ca6 [0, 0, 0, -7901915916, 270362650933744] [2] 47185920  
68544.bq2 68544ca4 [0, 0, 0, -494809356, 4207535177200] [2, 2] 23592960  
68544.bq3 68544ca5 [0, 0, 0, -168539916, 9673331343856] [2] 47185920  
68544.bq4 68544ca2 [0, 0, 0, -52257036, -36541571600] [2, 2] 11796480  
68544.bq5 68544ca1 [0, 0, 0, -40460556, -98935513616] [2] 5898240 \(\Gamma_0(N)\)-optimal
68544.bq6 68544ca3 [0, 0, 0, 201551604, -287406031376] [2] 23592960  

Rank

sage: E.rank()
 

The elliptic curves in class 68544.bq have rank \(1\).

Modular form 68544.2.a.bq

sage: E.q_eigenform(10)
 
\( q - 2q^{5} + q^{7} + 4q^{11} + 2q^{13} - q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.