Properties

Label 840f
Number of curves $6$
Conductor $840$
CM no
Rank $1$
Graph

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

Elliptic curves in class 840f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
840.d4 840f1 \([0, -1, 0, -175, 952]\) \(37256083456/525\) \(8400\) \([4]\) \(128\) \(-0.10681\) \(\Gamma_0(N)\)-optimal
840.d3 840f2 \([0, -1, 0, -180, 900]\) \(2533446736/275625\) \(70560000\) \([2, 4]\) \(256\) \(0.23977\)  
840.d2 840f3 \([0, -1, 0, -680, -5700]\) \(34008619684/4862025\) \(4978713600\) \([2, 2]\) \(512\) \(0.58634\)  
840.d5 840f4 \([0, -1, 0, 240, 4092]\) \(1486779836/8203125\) \(-8400000000\) \([4]\) \(512\) \(0.58634\)  
840.d1 840f5 \([0, -1, 0, -10480, -409460]\) \(62161150998242/1607445\) \(3292047360\) \([2]\) \(1024\) \(0.93291\)  
840.d6 840f6 \([0, -1, 0, 1120, -32340]\) \(75798394558/259416045\) \(-531284060160\) \([2]\) \(1024\) \(0.93291\)  

Rank

sage: E.rank()
 

The elliptic curves in class 840f have rank \(1\).

Complex multiplication

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

Modular form 840.2.a.f

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