Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 1, -14625, 680781]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 1, -14625, 680781]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 1, -14625, 680781]) 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 8325.bd1 has rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1\)
\(37\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - 2 T + 2 T^{2}\) 1.2.ac
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 - 7 T + 13 T^{2}\) 1.13.ah
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 8325.bd do not have complex multiplication.

Modular form 8325.2.a.bd

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);
 
\(q + 2 q^{2} + 2 q^{4} + q^{7} - 6 q^{11} + 7 q^{13} + 2 q^{14} - 4 q^{16} + 4 q^{17} + 5 q^{19} + O(q^{20})\) 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 8325.bd

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
8325.bd1 8325k1 \([0, 0, 1, -14625, 680781]\) \(-32801034240/1369\) \(-14438671875\) \([]\) \(23040\) \(1.0301\) \(\Gamma_0(N)\)-optimal