Properties

Label 6720bg
Number of curves 6
Conductor 6720
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("6720.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.k6 6720bg1 [0, -1, 0, 639, 8481] [2] 6144 \(\Gamma_0(N)\)-optimal
6720.k5 6720bg2 [0, -1, 0, -4481, 91425] [2, 2] 12288  
6720.k4 6720bg3 [0, -1, 0, -23681, -1317855] [2] 24576  
6720.k2 6720bg4 [0, -1, 0, -67201, 6727201] [2, 2] 24576  
6720.k1 6720bg5 [0, -1, 0, -1075201, 429482401] [2] 49152  
6720.k3 6720bg6 [0, -1, 0, -62721, 7658145] [2] 49152  

Rank

sage: E.rank()
 

The elliptic curves in class 6720bg have rank \(0\).

Modular form 6720.2.a.k

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 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.