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("j1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6720.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.j1 6720c7 \([0, -1, 0, -122931201, -524574745599]\) \(783736670177727068275201/360150\) \(94411161600\) \([2]\) \(393216\) \(2.8341\)  
6720.j2 6720c5 \([0, -1, 0, -7683201, -8194556799]\) \(191342053882402567201/129708022500\) \(34002179850240000\) \([2, 2]\) \(196608\) \(2.4876\)  
6720.j3 6720c8 \([0, -1, 0, -7635201, -8302047999]\) \(-187778242790732059201/4984939585440150\) \(-1306772002685622681600\) \([2]\) \(393216\) \(2.8341\)  
6720.j4 6720c4 \([0, -1, 0, -964481, 364797825]\) \(378499465220294881/120530818800\) \(31596430963507200\) \([2]\) \(98304\) \(2.1410\)  
6720.j5 6720c3 \([0, -1, 0, -483201, -126236799]\) \(47595748626367201/1215506250000\) \(318637670400000000\) \([2, 2]\) \(98304\) \(2.1410\)  
6720.j6 6720c2 \([0, -1, 0, -68481, 4068225]\) \(135487869158881/51438240000\) \(13484225986560000\) \([2, 2]\) \(49152\) \(1.7944\)  
6720.j7 6720c1 \([0, -1, 0, 13439, 447361]\) \(1023887723039/928972800\) \(-243524645683200\) \([2]\) \(24576\) \(1.4479\) \(\Gamma_0(N)\)-optimal
6720.j8 6720c6 \([0, -1, 0, 81279, -404073855]\) \(226523624554079/269165039062500\) \(-70560000000000000000\) \([2]\) \(196608\) \(2.4876\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 6720.2.a.j

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

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.