Properties

Label 6864.e
Number of curves $6$
Conductor $6864$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("6864.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6864.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6864.e1 6864r3 [0, -1, 0, -109824, -13972032] [2] 16384  
6864.e2 6864r5 [0, -1, 0, -48144, 3953280] [4] 32768  
6864.e3 6864r4 [0, -1, 0, -7584, -167616] [2, 4] 16384  
6864.e4 6864r2 [0, -1, 0, -6864, -216576] [2, 2] 8192  
6864.e5 6864r1 [0, -1, 0, -384, -4032] [2] 4096 \(\Gamma_0(N)\)-optimal
6864.e6 6864r6 [0, -1, 0, 21456, -1166592] [4] 32768  

Rank

sage: E.rank()
 

The elliptic curves in class 6864.e have rank \(0\).

Modular form 6864.2.a.e

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.