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Properties

Label 422331br
Number of curves $6$
Conductor $422331$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("br1")
 
E.isogeny_class()
 

Elliptic curves in class 422331br

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422331.br4 422331br1 \([1, 0, 1, -13614137, -19335646081]\) \(491411892194497/78897\) \(44803180378278777\) \([2]\) \(12386304\) \(2.5976\) \(\Gamma_0(N)\)-optimal*
422331.br3 422331br2 \([1, 0, 1, -13655542, -19212126685]\) \(495909170514577/6224736609\) \(3534836522305060668969\) \([2, 2]\) \(24772608\) \(2.9442\) \(\Gamma_0(N)\)-optimal*
422331.br2 422331br3 \([1, 0, 1, -25621587, 19533927025]\) \(3275619238041697/1605271262049\) \(911584190902677739092009\) \([2, 2]\) \(49545216\) \(3.2907\) \(\Gamma_0(N)\)-optimal*
422331.br5 422331br4 \([1, 0, 1, -2351977, -50052773431]\) \(-2533811507137/1904381781393\) \(-1081439857800150081073113\) \([2]\) \(49545216\) \(3.2907\)  
422331.br1 422331br5 \([1, 0, 1, -336034872, 2369238329161]\) \(7389727131216686257/6115533215337\) \(3472823272825313979752817\) \([2]\) \(99090432\) \(3.6373\) \(\Gamma_0(N)\)-optimal*
422331.br6 422331br6 \([1, 0, 1, 93334978, 149624826509]\) \(158346567380527343/108665074944153\) \(-61707554831515373956266273\) \([2]\) \(99090432\) \(3.6373\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 422331br1.

Rank

sage: E.rank()
 

The elliptic curves in class 422331br have rank \(1\).

Complex multiplication

The elliptic curves in class 422331br do not have complex multiplication.

Modular form 422331.2.a.br

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} - 2 q^{5} + q^{6} - 3 q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - 2 q^{15} - q^{16} - q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.