Properties

Label 4046b
Number of curves 6
Conductor 4046
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("4046.f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4046b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4046.f5 4046b1 [1, 1, 0, -150, 1376] [2] 1536 \(\Gamma_0(N)\)-optimal
4046.f4 4046b2 [1, 1, 0, -3040, 63222] [2] 3072  
4046.f6 4046b3 [1, 1, 0, 1295, -29547] [2] 4608  
4046.f3 4046b4 [1, 1, 0, -10265, -332419] [2] 9216  
4046.f2 4046b5 [1, 1, 0, -49280, -4243456] [2] 13824  
4046.f1 4046b6 [1, 1, 0, -789120, -270141952] [2] 27648  

Rank

sage: E.rank()
 

The elliptic curves in class 4046b have rank \(1\).

Modular form 4046.2.a.f

sage: E.q_eigenform(10)
 
\( q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - q^{7} - q^{8} + q^{9} + 2q^{12} - 4q^{13} + q^{14} + q^{16} - q^{18} + 2q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.