Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 1, -147, -86]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 1, -147, -86]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 1, -147, -86]) 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 1323r1 has rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - 2 T + 2 T^{2}\) 1.2.ac
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1323r do not have complex multiplication.

Modular form 1323.2.a.r

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);
 
\(q - 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{10} - 4 q^{11} + 2 q^{13} - 4 q^{16} - 3 q^{17} + 8 q^{19} + O(q^{20})\) 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 1323r

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
1323.b1 1323r1 \([0, 0, 1, -147, -86]\) \(12288/7\) \(200120949\) \([]\) \(576\) \(0.28269\) \(\Gamma_0(N)\)-optimal