Properties

Label 56784cn
Number of curves $6$
Conductor $56784$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("56784.cq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 56784cn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
56784.cq5 56784cn1 [0, 1, 0, -10872, -967212] [2] 184320 \(\Gamma_0(N)\)-optimal
56784.cq4 56784cn2 [0, 1, 0, -227192, -41721900] [2, 2] 368640  
56784.cq3 56784cn3 [0, 1, 0, -281272, -20414380] [2, 2] 737280  
56784.cq1 56784cn4 [0, 1, 0, -3634232, -2667868332] [2] 737280  
56784.cq6 56784cn5 [0, 1, 0, 1043688, -156620268] [2] 1474560  
56784.cq2 56784cn6 [0, 1, 0, -2471512, 1480338068] [2] 1474560  

Rank

sage: E.rank()
 

The elliptic curves in class 56784cn have rank \(1\).

Modular form 56784.2.a.cq

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.