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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -5321216705, 193788826336575]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -5321216705, 193788826336575]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -5321216705, 193788826336575]) 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 46090.n1 has rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
\(419\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 - T + 23 T^{2}\) 1.23.ab
\(29\) \( 1 - 5 T + 29 T^{2}\) 1.29.af
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 46090.n do not have complex multiplication.

Modular form 46090.2.a.n

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);
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9} + q^{10} + q^{11} - q^{12} + q^{13} + 3 q^{14} - q^{15} + q^{16} - 2 q^{18} - 2 q^{19} + O(q^{20})\) 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 46090.n

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
46090.n1 46090o1 \([1, 1, 1, -5321216705, 193788826336575]\) \(-16663128733355465781481076750866321/6580732098057570116451334553600\) \(-6580732098057570116451334553600\) \([]\) \(70791168\) \(4.6225\) \(\Gamma_0(N)\)-optimal