Properties

Label 8400.p
Number of curves $6$
Conductor $8400$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("8400.p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8400.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8400.p1 8400bp5 [0, -1, 0, -6720008, 6707302512] [2] 147456  
8400.p2 8400bp4 [0, -1, 0, -420008, 104902512] [2, 2] 73728  
8400.p3 8400bp6 [0, -1, 0, -392008, 119462512] [2] 147456  
8400.p4 8400bp3 [0, -1, 0, -148008, -20665488] [2] 73728  
8400.p5 8400bp2 [0, -1, 0, -28008, 1414512] [2, 2] 36864  
8400.p6 8400bp1 [0, -1, 0, 3992, 134512] [2] 18432 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8400.p have rank \(1\).

Modular form 8400.2.a.p

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.