↑

Properties

Label 223850bo
Number of curves $4$
Conductor $223850$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -226938, 22657031]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -226938, 22657031]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -226938, 22657031]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 223850bo have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(5\)\(1\)
\(11\)\(1\)
\(37\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 223850bo do not have complex multiplication.

Modular form 223850.2.a.bo

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} + 2 q^{3} + q^{4} + 2 q^{6} + 2 q^{7} + q^{8} + q^{9} + 2 q^{12} + 2 q^{13} + 2 q^{14} + q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 223850bo

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
223850.dn3 223850bo1 \([1, 1, 1, -226938, 22657031]\) \(46694890801/18944000\) \(524382056000000000\) \([2]\) \(3317760\) \(2.0979\) \(\Gamma_0(N)\)-optimal
223850.dn4 223850bo2 \([1, 1, 1, 741062, 165921031]\) \(1625964918479/1369000000\) \(-37894797015625000000\) \([2]\) \(6635520\) \(2.4444\)  
223850.dn1 223850bo3 \([1, 1, 1, -15956938, 24527577031]\) \(16232905099479601/4052240\) \(112168599166250000\) \([2]\) \(9953280\) \(2.6472\)  
223850.dn2 223850bo4 \([1, 1, 1, -15896438, 24722871031]\) \(-16048965315233521/256572640900\) \(-7102095066960076562500\) \([2]\) \(19906560\) \(2.9937\)