Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, -14572, 855033]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -14572, 855033]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -14572, 855033]) 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 39326.k1 has rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(7\)\(1 + T\)
\(53\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 9 T + 23 T^{2}\) 1.23.aj
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 39326.k do not have complex multiplication.

Modular form 39326.2.a.k

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);
 
\(q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} - 3 q^{9} + q^{10} + 3 q^{11} + 4 q^{13} - q^{14} + q^{16} - 3 q^{17} - 3 q^{18} + O(q^{20})\) 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 39326.k

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
39326.k1 39326f1 \([1, -1, 1, -14572, 855033]\) \(-15438249/5194\) \(-115121691704026\) \([]\) \(112320\) \(1.4100\) \(\Gamma_0(N)\)-optimal