Properties

Label 49686.bd
Number of curves $6$
Conductor $49686$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("49686.bd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 49686.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
49686.bd1 49686bk4 [1, 0, 1, -11129837, -14292541924] [2] 1474560  
49686.bd2 49686bk6 [1, 0, 1, -7569007, 7937471336] [2] 2949120  
49686.bd3 49686bk3 [1, 0, 1, -861397, -108977620] [2, 2] 1474560  
49686.bd4 49686bk2 [1, 0, 1, -695777, -223255420] [2, 2] 737280  
49686.bd5 49686bk1 [1, 0, 1, -33297, -5167004] [2] 368640 \(\Gamma_0(N)\)-optimal
49686.bd6 49686bk5 [1, 0, 1, 3196293, -840984896] [2] 2949120  

Rank

sage: E.rank()
 

The elliptic curves in class 49686.bd have rank \(1\).

Modular form 49686.2.a.bd

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.