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Properties

Label 80850.bs
Number of curves $6$
Conductor $80850$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bs1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 80850.bs have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 80850.bs do not have complex multiplication.

Modular form 80850.2.a.bs

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + q^{11} - q^{12} + 6 q^{13} + q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 80850.bs

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80850.bs1 80850w6 \([1, 1, 0, -209579150, 1167717375000]\) \(553808571467029327441/12529687500\) \(23032893823242187500\) \([2]\) \(14155776\) \(3.2394\)  
80850.bs2 80850w4 \([1, 1, 0, -14485650, -21180037500]\) \(182864522286982801/463015182960\) \(851144894688453750000\) \([2]\) \(7077888\) \(2.8929\)  
80850.bs3 80850w3 \([1, 1, 0, -13113650, 18197734500]\) \(135670761487282321/643043610000\) \(1182084963638906250000\) \([2, 2]\) \(7077888\) \(2.8929\)  
80850.bs4 80850w5 \([1, 1, 0, -6376150, 36880822000]\) \(-15595206456730321/310672490129100\) \(-571098559237476342187500\) \([2]\) \(14155776\) \(3.2394\)  
80850.bs5 80850w2 \([1, 1, 0, -1255650, -51727500]\) \(119102750067601/68309049600\) \(125570177756100000000\) \([2, 2]\) \(3538944\) \(2.5463\)  
80850.bs6 80850w1 \([1, 1, 0, 312350, -6255500]\) \(1833318007919/1070530560\) \(-1967919528960000000\) \([2]\) \(1769472\) \(2.1997\) \(\Gamma_0(N)\)-optimal