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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -5708, 104912]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -5708, 104912]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -5708, 104912]) 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 4400.x have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(5\)\(1\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 + 13 T^{2}\) 1.13.a
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 4400.x do not have complex multiplication.

Modular form 4400.2.a.x

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);
 
\(q + 2 q^{3} - 2 q^{7} + q^{9} + q^{11} - 4 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

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 4400.x

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
4400.x1 4400j2 \([0, -1, 0, -5708, 104912]\) \(41141648/14641\) \(7320500000000\) \([2]\) \(7680\) \(1.1695\)  
4400.x2 4400j1 \([0, -1, 0, -5083, 141162]\) \(464857088/121\) \(3781250000\) \([2]\) \(3840\) \(0.82291\) \(\Gamma_0(N)\)-optimal