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Properties

Label 85176.n
Number of curves $6$
Conductor $85176$
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, 0, 0, -4305136251, 108724761767414]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -4305136251, 108724761767414]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -4305136251, 108724761767414]) 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 85176.n have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 85176.n do not have complex multiplication.

Modular form 85176.2.a.n

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);
 
\(q - 2 q^{5} - q^{7} - 4 q^{11} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

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 85176.n

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
85176.n1 85176bw6 \([0, 0, 0, -4305136251, 108724761767414]\) \(1224522642327678150914/66339\) \(478064521955284992\) \([2]\) \(24772608\) \(3.7831\)  
85176.n2 85176bw4 \([0, 0, 0, -269071491, 1698818100590]\) \(597914615076708388/4400862921\) \(15857161160995825542144\) \([2, 2]\) \(12386304\) \(3.4365\)  
85176.n3 85176bw5 \([0, 0, 0, -263535051, 1772072952806]\) \(-280880296871140514/25701087819771\) \(-185211990869467747735001088\) \([2]\) \(24772608\) \(3.7831\)  
85176.n4 85176bw3 \([0, 0, 0, -57409131, -137947911946]\) \(5807363790481348/1079211743883\) \(3888608860760305831898112\) \([2]\) \(12386304\) \(3.4365\)  
85176.n5 85176bw2 \([0, 0, 0, -17163471, 25393123730]\) \(620742479063632/49991146569\) \(45031944984295758338304\) \([2, 2]\) \(6193152\) \(3.0899\)  
85176.n6 85176bw1 \([0, 0, 0, 1096134, 1798062149]\) \(2587063175168/26304786963\) \(-1480956880167846205488\) \([2]\) \(3096576\) \(2.7433\) \(\Gamma_0(N)\)-optimal