Properties

Label 8664.j
Number of curves $6$
Conductor $8664$
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 8664.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8664.j1 8664g5 \([0, 1, 0, -138744, 19845360]\) \(3065617154/9\) \(867149678592\) \([2]\) \(27648\) \(1.5200\)  
8664.j2 8664g3 \([0, 1, 0, -23224, -1369888]\) \(28756228/3\) \(144524946432\) \([2]\) \(13824\) \(1.1734\)  
8664.j3 8664g4 \([0, 1, 0, -8784, 299376]\) \(1556068/81\) \(3902173553664\) \([2, 2]\) \(13824\) \(1.1734\)  
8664.j4 8664g2 \([0, 1, 0, -1564, -18304]\) \(35152/9\) \(108393709824\) \([2, 2]\) \(6912\) \(0.82687\)  
8664.j5 8664g1 \([0, 1, 0, 241, -1698]\) \(2048/3\) \(-2258202288\) \([2]\) \(3456\) \(0.48029\) \(\Gamma_0(N)\)-optimal
8664.j6 8664g6 \([0, 1, 0, 5656, 1200432]\) \(207646/6561\) \(-632152115693568\) \([2]\) \(27648\) \(1.5200\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8664.j have rank \(1\).

Complex multiplication

The elliptic curves in class 8664.j do not have complex multiplication.

Modular form 8664.2.a.j

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.