Properties

Label 6720.cg
Number of curves $4$
Conductor $6720$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("cg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.cg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.cg1 6720y3 [0, 1, 0, -23905, 1414655] [4] 12288  
6720.cg2 6720y2 [0, 1, 0, -1505, 21375] [2, 2] 6144  
6720.cg3 6720y1 [0, 1, 0, -225, -897] [2] 3072 \(\Gamma_0(N)\)-optimal
6720.cg4 6720y4 [0, 1, 0, 415, 73983] [2] 12288  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 6720.cg do not have complex multiplication.

Modular form 6720.2.a.cg

sage: E.q_eigenform(10)
 
\( q + q^{3} + q^{5} - q^{7} + q^{9} + 4q^{11} + 2q^{13} + q^{15} - 6q^{17} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.