Properties

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

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("6720.bp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.bp6 6720u1 [0, 1, 0, 639, -8481] [2] 6144 \(\Gamma_0(N)\)-optimal
6720.bp5 6720u2 [0, 1, 0, -4481, -91425] [2, 2] 12288  
6720.bp2 6720u3 [0, 1, 0, -67201, -6727201] [2, 2] 24576  
6720.bp4 6720u4 [0, 1, 0, -23681, 1317855] [2] 24576  
6720.bp1 6720u5 [0, 1, 0, -1075201, -429482401] [2] 49152  
6720.bp3 6720u6 [0, 1, 0, -62721, -7658145] [2] 49152  

Rank

sage: E.rank()
 

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

Modular form 6720.2.a.bp

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.