Properties

Label 6720.j
Number of curves 8
Conductor 6720
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("6720.j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.j1 6720c7 [0, -1, 0, -122931201, -524574745599] [2] 393216  
6720.j2 6720c5 [0, -1, 0, -7683201, -8194556799] [2, 2] 196608  
6720.j3 6720c8 [0, -1, 0, -7635201, -8302047999] [2] 393216  
6720.j4 6720c4 [0, -1, 0, -964481, 364797825] [2] 98304  
6720.j5 6720c3 [0, -1, 0, -483201, -126236799] [2, 2] 98304  
6720.j6 6720c2 [0, -1, 0, -68481, 4068225] [2, 2] 49152  
6720.j7 6720c1 [0, -1, 0, 13439, 447361] [2] 24576 \(\Gamma_0(N)\)-optimal
6720.j8 6720c6 [0, -1, 0, 81279, -404073855] [2] 196608  

Rank

sage: E.rank()
 

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

Modular form 6720.2.a.j

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.