↑

Properties

Label 88806p
Number of curves $2$
Conductor $88806$
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, 1, -163856644, 807248887301]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -163856644, 807248887301]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -163856644, 807248887301]) 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 88806p have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(19\)\(1\)
\(41\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 + 13 T^{2}\) 1.13.a
\(17\) \( 1 - 8 T + 17 T^{2}\) 1.17.ai
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 88806p do not have complex multiplication.

Modular form 88806.2.a.p

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);
 
\(q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + 2 q^{7} + q^{8} + q^{9} - 2 q^{10} + 4 q^{11} - q^{12} - 4 q^{13} + 2 q^{14} + 2 q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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 88806p

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
88806.o1 88806p1 \([1, 1, 1, -163856644, 807248887301]\) \(10341755683137709164937/356992303104\) \(16795017409746714624\) \([2]\) \(12096000\) \(3.1866\) \(\Gamma_0(N)\)-optimal
88806.o2 88806p2 \([1, 1, 1, -163625604, 809639134725]\) \(-10298071306410575356297/60769798505543808\) \(-2858968708885791831454848\) \([2]\) \(24192000\) \(3.5332\)