Properties

Label 209814bv
Number of curves $4$
Conductor $209814$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bv1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 209814bv have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(11\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 209814bv do not have complex multiplication.

Modular form 209814.2.a.bv

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} + 4 q^{7} - q^{8} + q^{9} + 2 q^{10} + q^{12} + 2 q^{13} - 4 q^{14} - 2 q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 209814bv

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
209814.bl3 209814bv1 \([1, 0, 1, -13533732, -19163845934]\) \(6411014266033/296208\) \(12666202383643907472\) \([2]\) \(13271040\) \(2.7407\) \(\Gamma_0(N)\)-optimal
209814.bl2 209814bv2 \([1, 0, 1, -14233112, -17073538990]\) \(7457162887153/1370924676\) \(58622351182099914757284\) \([2, 2]\) \(26542080\) \(3.0872\)  
209814.bl1 209814bv3 \([1, 0, 1, -67735682, 198862833530]\) \(803760366578833/65593817586\) \(2804868770121323045625474\) \([2]\) \(53084160\) \(3.4338\)  
209814.bl4 209814bv4 \([1, 0, 1, 28079378, -99227469574]\) \(57258048889007/132611470002\) \(-5670622391825524398980418\) \([2]\) \(53084160\) \(3.4338\)