Properties

Label 720.j
Number of curves $8$
Conductor $720$
CM no
Rank $0$
Graph

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

Elliptic curves in class 720.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
720.j1 720j7 \([0, 0, 0, -768027, -259067446]\) \(16778985534208729/81000\) \(241864704000\) \([2]\) \(4608\) \(1.8066\)  
720.j2 720j8 \([0, 0, 0, -65307, -874294]\) \(10316097499609/5859375000\) \(17496000000000000\) \([4]\) \(4608\) \(1.8066\)  
720.j3 720j6 \([0, 0, 0, -48027, -4043446]\) \(4102915888729/9000000\) \(26873856000000\) \([2, 2]\) \(2304\) \(1.4600\)  
720.j4 720j5 \([0, 0, 0, -41547, 3259514]\) \(2656166199049/33750\) \(100776960000\) \([4]\) \(1536\) \(1.2573\)  
720.j5 720j4 \([0, 0, 0, -9867, -324934]\) \(35578826569/5314410\) \(15868743229440\) \([2]\) \(1536\) \(1.2573\)  
720.j6 720j2 \([0, 0, 0, -2667, 48026]\) \(702595369/72900\) \(217678233600\) \([2, 2]\) \(768\) \(0.91074\)  
720.j7 720j3 \([0, 0, 0, -1947, -108214]\) \(-273359449/1536000\) \(-4586471424000\) \([2]\) \(1152\) \(1.1135\)  
720.j8 720j1 \([0, 0, 0, 213, 3674]\) \(357911/2160\) \(-6449725440\) \([2]\) \(384\) \(0.56417\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 720.j have rank \(0\).

Complex multiplication

The elliptic curves in class 720.j do not have complex multiplication.

Modular form 720.2.a.j

sage: E.q_eigenform(10)
 
\(q + q^{5} + 4q^{7} + 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.