Properties

Label 840e
Number of curves $4$
Conductor $840$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 840e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
840.b4 840e1 \([0, -1, 0, 9, 0]\) \(4499456/2835\) \(-45360\) \([2]\) \(64\) \(-0.41743\) \(\Gamma_0(N)\)-optimal
840.b3 840e2 \([0, -1, 0, -36, 36]\) \(20720464/11025\) \(2822400\) \([2, 2]\) \(128\) \(-0.070861\)  
840.b2 840e3 \([0, -1, 0, -336, -2244]\) \(4108974916/36015\) \(36879360\) \([2]\) \(256\) \(0.27571\)  
840.b1 840e4 \([0, -1, 0, -456, 3900]\) \(10262905636/13125\) \(13440000\) \([2]\) \(256\) \(0.27571\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 840.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.