Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, 4116, -6953296]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 4116, -6953296]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 4116, -6953296]) 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 28224f have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 4 T + 5 T^{2}\) 1.5.e
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 8 T + 29 T^{2}\) 1.29.i
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 28224f do not have complex multiplication.

Modular form 28224.2.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);
 
\(q - 3 q^{5} + 3 q^{11} - 2 q^{13} + 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

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 28224f

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
28224.v2 28224f1 \([0, 0, 0, 4116, -6953296]\) \(189/512\) \(-20890939299987456\) \([]\) \(193536\) \(1.8105\) \(\Gamma_0(N)\)-optimal
28224.v1 28224f2 \([0, 0, 0, -2630124, -1641938256]\) \(-67645179/8\) \(-237960855463919616\) \([]\) \(580608\) \(2.3598\)