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Properties

Label 4400.s
Number of curves $3$
Conductor $4400$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -485248, -130268492]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -485248, -130268492]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -485248, -130268492]) 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.s 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 \(\mathbb{F}_p\)
\(3\) \( 1 - T + 3 T^{2}\) 1.3.ab
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 5 T + 29 T^{2}\) 1.29.af
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 4400.2.a.s

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);
 
\(q + q^{3} - 3 q^{7} - 2 q^{9} - q^{11} + 4 q^{13} + 3 q^{17} + 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 25 & 5 \\ 25 & 1 & 5 \\ 5 & 5 & 1 \end{array}\right)\)

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 4400.s

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.s1 4400v3 \([0, 1, 0, -485248, -130268492]\) \(-24680042791780949/369098752\) \(-188978561024000\) \([]\) \(28800\) \(1.8763\)  
4400.s2 4400v1 \([0, 1, 0, -448, 3508]\) \(-19465109/22\) \(-11264000\) \([]\) \(1152\) \(0.26684\) \(\Gamma_0(N)\)-optimal
4400.s3 4400v2 \([0, 1, 0, 3152, -37292]\) \(6761990971/5153632\) \(-2638659584000\) \([]\) \(5760\) \(1.0716\)