Properties

Label 177744.dd
Number of curves $6$
Conductor $177744$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("177744.dd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 177744.dd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177744.dd1 177744x6 [0, 1, 0, -669790352, -6672162455532] [2] 51904512  
177744.dd2 177744x3 [0, 1, 0, -149931472, 706553332628] [4] 25952256  
177744.dd3 177744x4 [0, 1, 0, -42946512, -98576474220] [2, 2] 25952256  
177744.dd4 177744x2 [0, 1, 0, -9767632, 10051178900] [2, 2] 12976128  
177744.dd5 177744x1 [0, 1, 0, 1066288, 868348308] [2] 6488064 \(\Gamma_0(N)\)-optimal
177744.dd6 177744x5 [0, 1, 0, 53035248, -476629430508] [2] 51904512  

Rank

sage: E.rank()
 

The elliptic curves in class 177744.dd have rank \(1\).

Modular form 177744.2.a.dd

sage: E.q_eigenform(10)
 
\( q + q^{3} + 2q^{5} + q^{7} + q^{9} - 4q^{11} - 2q^{13} + 2q^{15} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.