Properties

Label 476850.bk
Number of curves $6$
Conductor $476850$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bk1")
 
E.isogeny_class()
 

Elliptic curves in class 476850.bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
476850.bk1 476850bk6 \([1, 1, 0, -1236089275, -16727700137375]\) \(553808571467029327441/12529687500\) \(4725565571557617187500\) \([2]\) \(188743680\) \(3.6831\)  
476850.bk2 476850bk3 \([1, 1, 0, -85435775, 303252655125]\) \(182864522286982801/463015182960\) \(174625951980384753750000\) \([2]\) \(94371840\) \(3.3365\) \(\Gamma_0(N)\)-optimal*
476850.bk3 476850bk4 \([1, 1, 0, -77343775, -260767836875]\) \(135670761487282321/643043610000\) \(242523586037251406250000\) \([2, 2]\) \(94371840\) \(3.3365\)  
476850.bk4 476850bk5 \([1, 1, 0, -37606275, -528320424375]\) \(-15595206456730321/310672490129100\) \(-117169979170202658717187500\) \([2]\) \(188743680\) \(3.6831\)  
476850.bk5 476850bk2 \([1, 1, 0, -7405775, 730345125]\) \(119102750067601/68309049600\) \(25762724969444100000000\) \([2, 2]\) \(47185920\) \(2.9899\) \(\Gamma_0(N)\)-optimal*
476850.bk6 476850bk1 \([1, 1, 0, 1842225, 92233125]\) \(1833318007919/1070530560\) \(-403750082165760000000\) \([2]\) \(23592960\) \(2.6434\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 476850.bk1.

Rank

sage: E.rank()
 

The elliptic curves in class 476850.bk have rank \(1\).

Complex multiplication

The elliptic curves in class 476850.bk do not have complex multiplication.

Modular form 476850.2.a.bk

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{11} - q^{12} - 6 q^{13} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.