↑

Properties

Label 54600q
Number of curves $2$
Conductor $54600$
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([0, -1, 0, -2667068, 1677368532]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -2667068, 1677368532]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -2667068, 1677368532]) 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 54600q have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 5 T + 23 T^{2}\) 1.23.af
\(29\) \( 1 + 9 T + 29 T^{2}\) 1.29.j
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 54600q do not have complex multiplication.

Modular form 54600.2.a.q

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);
 
\(q - q^{3} + q^{7} + q^{9} - 4 q^{11} - q^{13} + 6 q^{17} + 2 q^{19} + O(q^{20})\) 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.

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

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 54600q

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
54600.x1 54600q1 \([0, -1, 0, -2667068, 1677368532]\) \(65565618540844760336/188428167837\) \(6029701370784000\) \([2]\) \(940032\) \(2.2585\) \(\Gamma_0(N)\)-optimal
54600.x2 54600q2 \([0, -1, 0, -2632768, 1722575932]\) \(-15767094823546327124/879851321206767\) \(-112620969114466176000\) \([2]\) \(1880064\) \(2.6051\)