Properties

Label 840j
Number of curves $6$
Conductor $840$
CM no
Rank $0$
Graph

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

Elliptic curves in class 840j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
840.j5 840j1 \([0, 1, 0, -15, 90]\) \(-24918016/229635\) \(-3674160\) \([4]\) \(128\) \(-0.057811\) \(\Gamma_0(N)\)-optimal
840.j4 840j2 \([0, 1, 0, -420, 3168]\) \(32082281296/99225\) \(25401600\) \([2, 4]\) \(256\) \(0.28876\)  
840.j3 840j3 \([0, 1, 0, -600, 0]\) \(23366901604/13505625\) \(13829760000\) \([2, 2]\) \(512\) \(0.63534\)  
840.j1 840j4 \([0, 1, 0, -6720, 209808]\) \(32779037733124/315\) \(322560\) \([4]\) \(512\) \(0.63534\)  
840.j2 840j5 \([0, 1, 0, -6480, -202272]\) \(14695548366242/57421875\) \(117600000000\) \([2]\) \(1024\) \(0.98191\)  
840.j6 840j6 \([0, 1, 0, 2400, 2400]\) \(746185003198/432360075\) \(-885473433600\) \([2]\) \(1024\) \(0.98191\)  

Rank

sage: E.rank()
 

The elliptic curves in class 840j have rank \(0\).

Complex multiplication

The elliptic curves in class 840j do not have complex multiplication.

Modular form 840.2.a.j

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