Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -9, -8]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -9, -8]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -9, -8]) 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 curve 10388.b1 has rank \(2\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1\)
\(53\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 + T + 17 T^{2}\) 1.17.b
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 9 T + 29 T^{2}\) 1.29.j
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 10388.b do not have complex multiplication.

Modular form 10388.2.a.b

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);
 
\(q - 2 q^{3} - 2 q^{5} + q^{9} - 3 q^{11} + q^{13} + 4 q^{15} - q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny graph

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

Elliptic curves in class 10388.b

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
10388.b1 10388i1 \([0, 1, 0, -9, -8]\) \(114688/53\) \(41552\) \([]\) \(792\) \(-0.41911\) \(\Gamma_0(N)\)-optimal