↑

Properties

Label 5424l
Number of curves $2$
Conductor $5424$
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, -26576, -1670208]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -26576, -1670208]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -26576, -1670208]) 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 5424l have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(113\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 7 T + 23 T^{2}\) 1.23.ah
\(29\) \( 1 + T + 29 T^{2}\) 1.29.b
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 5424l do not have complex multiplication.

Modular form 5424.2.a.l

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);
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{11} + 7 q^{13} + q^{15} - 3 q^{17} - 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

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 5424l

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
5424.e2 5424l1 \([0, -1, 0, -26576, -1670208]\) \(-506814405937489/4048994304\) \(-16584680669184\) \([]\) \(12096\) \(1.3646\) \(\Gamma_0(N)\)-optimal
5424.e1 5424l2 \([0, -1, 0, -113936, 164313792]\) \(-39934705050538129/2823126576537804\) \(-11563526457498845184\) \([]\) \(84672\) \(2.3376\)