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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, 13986, 1058640]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, 13986, 1058640]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, 13986, 1058640]) 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 curves in class 162922f have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(29\)\(1 + T\)
\(53\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 2 T + 23 T^{2}\) 1.23.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 162922f do not have complex multiplication.

Modular form 162922.2.a.f

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);
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 2 q^{7} - q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - q^{13} + 2 q^{14} - q^{15} + q^{16} + 8 q^{17} + 2 q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labeled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 162922f

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
162922.b2 162922f1 \([1, 0, 1, 13986, 1058640]\) \(13651919/29696\) \(-658192868086784\) \([]\) \(581360\) \(1.5276\) \(\Gamma_0(N)\)-optimal
162922.b1 162922f2 \([1, 0, 1, -1278154, -560067200]\) \(-10418796526321/82044596\) \(-1818466054426908884\) \([]\) \(2906800\) \(2.3323\)