Properties

Label 55440dr
Number of curves $6$
Conductor $55440$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("55440.cf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 55440dr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55440.cf4 55440dr1 [0, 0, 0, -38163, 2869522] [2] 131072 \(\Gamma_0(N)\)-optimal
55440.cf3 55440dr2 [0, 0, 0, -38883, 2755618] [2, 2] 262144  
55440.cf5 55440dr3 [0, 0, 0, 36717, 12175378] [2] 524288  
55440.cf2 55440dr4 [0, 0, 0, -126003, -13953998] [2, 2] 524288  
55440.cf6 55440dr5 [0, 0, 0, 262077, -82954622] [2] 1048576  
55440.cf1 55440dr6 [0, 0, 0, -1908003, -1014368798] [2] 1048576  

Rank

sage: E.rank()
 

The elliptic curves in class 55440dr have rank \(1\).

Modular form 55440.2.a.cf

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

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 Cremona 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 Cremona labels.