↑

Properties

Label 15210o
Number of curves $4$
Conductor $15210$
CM no
Rank $0$
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, 0, -438840, -184745664]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -438840, -184745664]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -438840, -184745664]) 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 15210o have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 15210o do not have complex multiplication.

Modular form 15210.2.a.o

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);
 
\(q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8} + q^{10} + 4 q^{14} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 15210o

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
15210.a4 15210o1 \([1, -1, 0, -438840, -184745664]\) \(-2656166199049/2658140160\) \(-9353314103863541760\) \([2]\) \(430080\) \(2.3359\) \(\Gamma_0(N)\)-optimal
15210.a3 15210o2 \([1, -1, 0, -8226360, -9076536000]\) \(17496824387403529/6580454400\) \(23154932864544998400\) \([2, 2]\) \(860160\) \(2.6825\)  
15210.a1 15210o3 \([1, -1, 0, -131609880, -581107211424]\) \(71647584155243142409/10140000\) \(35680061736540000\) \([2]\) \(1720320\) \(3.0290\)  
15210.a2 15210o4 \([1, -1, 0, -9443160, -6213892320]\) \(26465989780414729/10571870144160\) \(37199702111865170585760\) \([2]\) \(1720320\) \(3.0290\)