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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, 51, 117]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, 51, 117]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, 51, 117]) 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 974.e1 has rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(487\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T + 3 T^{2}\) 1.3.d
\(5\) \( 1 + 3 T + 5 T^{2}\) 1.5.d
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 974.e do not have complex multiplication.

Modular form 974.2.a.e

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);
 
\(q + q^{2} - 3 q^{3} + q^{4} - 3 q^{5} - 3 q^{6} + 2 q^{7} + q^{8} + 6 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} - 2 q^{13} + 2 q^{14} + 9 q^{15} + q^{16} + 2 q^{17} + 6 q^{18} - 6 q^{19} + O(q^{20})\) 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 974.e

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
974.e1 974h1 \([1, -1, 1, 51, 117]\) \(14941466847/15958016\) \(-15958016\) \([]\) \(480\) \(0.068780\) \(\Gamma_0(N)\)-optimal