Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -54421, 4945517]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -54421, 4945517]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -54421, 4945517]) 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 338.b have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 338.b do not have complex multiplication.

Modular form 338.2.a.b

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);
 
\(q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} - q^{8} - 2 q^{9} - 3 q^{10} - q^{12} - 3 q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 2 q^{18} + 6 q^{19} + O(q^{20})\) 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 LMFDB 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 labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 338.b

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
338.b1 338d2 \([1, 1, 0, -54421, 4945517]\) \(-1680914269/32768\) \(-347488235454464\) \([]\) \(1560\) \(1.5836\)  
338.b2 338d1 \([1, 1, 0, 504, -13112]\) \(1331/8\) \(-84835994984\) \([]\) \(312\) \(0.77890\) \(\Gamma_0(N)\)-optimal