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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -2226316, -1477144936]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -2226316, -1477144936]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -2226316, -1477144936]) 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 64974bx have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(13\)\(1 + T\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 + 3 T + 23 T^{2}\) 1.23.d
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 64974bx do not have complex multiplication.

Modular form 64974.2.a.bx

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 64974bx

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
64974.bw2 64974bx1 \([1, 0, 0, -2226316, -1477144936]\) \(-24905087205614147556241/4819348696095929736\) \(-236148086108700557064\) \([]\) \(2088576\) \(2.6328\) \(\Gamma_0(N)\)-optimal
64974.bw1 64974bx2 \([1, 0, 0, -87674406, 422770263012]\) \(-1521059241134755603512440881/695595284594977727840256\) \(-34084168945153908664172544\) \([7]\) \(14620032\) \(3.6057\)