Properties

Label 6720.m
Number of curves $6$
Conductor $6720$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("6720.m1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.m1 6720bm5 [0, -1, 0, -41921, 3317601] [2] 16384  
6720.m2 6720bm3 [0, -1, 0, -2721, 48321] [2, 2] 8192  
6720.m3 6720bm2 [0, -1, 0, -721, -6479] [2, 2] 4096  
6720.m4 6720bm1 [0, -1, 0, -701, -6915] [2] 2048 \(\Gamma_0(N)\)-optimal
6720.m5 6720bm4 [0, -1, 0, 959, -33695] [2] 8192  
6720.m6 6720bm6 [0, -1, 0, 4479, 254241] [2] 16384  

Rank

sage: E.rank()
 

The elliptic curves in class 6720.m have rank \(1\).

Modular form 6720.2.a.m

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.