Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 1, 2113, -29546]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, 2113, -29546]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, 2113, -29546]) 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 143055f have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1 + T\)
\(11\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - T + 19 T^{2}\) 1.19.ab
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 9 T + 29 T^{2}\) 1.29.aj
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 143055f do not have complex multiplication.

Modular form 143055.2.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);
 
\(q - q^{2} - q^{4} + q^{5} + 3 q^{8} - q^{10} - q^{11} + 2 q^{13} - q^{16} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

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 143055f

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
143055.m4 143055f1 \([1, -1, 1, 2113, -29546]\) \(59319/55\) \(-967795829055\) \([2]\) \(147456\) \(0.98710\) \(\Gamma_0(N)\)-optimal
143055.m3 143055f2 \([1, -1, 1, -10892, -258434]\) \(8120601/3025\) \(53228770598025\) \([2, 2]\) \(294912\) \(1.3337\)  
143055.m2 143055f3 \([1, -1, 1, -75917, 7882696]\) \(2749884201/73205\) \(1288136248472205\) \([2]\) \(589824\) \(1.6802\)  
143055.m1 143055f4 \([1, -1, 1, -153947, -23204456]\) \(22930509321/6875\) \(120974478631875\) \([2]\) \(589824\) \(1.6802\)