Properties

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

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 840.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
840.f1 840g5 \([0, -1, 0, -24400, -1458548]\) \(784478485879202/221484375\) \(453600000000\) \([2]\) \(2048\) \(1.2177\)  
840.f2 840g3 \([0, -1, 0, -1720, -16100]\) \(549871953124/200930625\) \(205752960000\) \([2, 2]\) \(1024\) \(0.87118\)  
840.f3 840g2 \([0, -1, 0, -740, 7812]\) \(175293437776/4862025\) \(1244678400\) \([2, 4]\) \(512\) \(0.52460\)  
840.f4 840g1 \([0, -1, 0, -735, 7920]\) \(2748251600896/2205\) \(35280\) \([4]\) \(256\) \(0.17803\) \(\Gamma_0(N)\)-optimal
840.f5 840g4 \([0, -1, 0, 160, 24732]\) \(439608956/259416045\) \(-265642030080\) \([4]\) \(1024\) \(0.87118\)  
840.f6 840g6 \([0, -1, 0, 5280, -119700]\) \(7947184069438/7533176175\) \(-15427944806400\) \([2]\) \(2048\) \(1.2177\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 840.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{7} + q^{9} + 4 q^{11} - 2 q^{13} - q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.