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Properties

Label 4160.h
Number of curves $4$
Conductor $4160$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -88748, 10176208]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -88748, 10176208]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -88748, 10176208]) 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 4160.h have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(5\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 4160.h do not have complex multiplication.

Modular form 4160.2.a.h

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);
 
\(q - q^{5} - 3 q^{9} - q^{13} + 2 q^{17} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 4160.h

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
4160.h1 4160j3 \([0, 0, 0, -88748, 10176208]\) \(294889639316481/260\) \(68157440\) \([2]\) \(6144\) \(1.2355\)  
4160.h2 4160j2 \([0, 0, 0, -5548, 158928]\) \(72043225281/67600\) \(17720934400\) \([2, 2]\) \(3072\) \(0.88893\)  
4160.h3 4160j4 \([0, 0, 0, -4268, 234192]\) \(-32798729601/71402500\) \(-18717736960000\) \([2]\) \(6144\) \(1.2355\)  
4160.h4 4160j1 \([0, 0, 0, -428, 1232]\) \(33076161/16640\) \(4362076160\) \([2]\) \(1536\) \(0.54236\) \(\Gamma_0(N)\)-optimal