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Properties

Label 187200.hh
Number of curves $6$
Conductor $187200$
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, -129816300, -569300942000]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -129816300, -569300942000]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -129816300, -569300942000]) 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 187200.hh have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 187200.hh do not have complex multiplication.

Modular form 187200.2.a.hh

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);
 
\(q - 4 q^{11} + q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 187200.hh

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
187200.hh1 187200ed6 \([0, 0, 0, -129816300, -569300942000]\) \(81025909800741361/11088090\) \(33108859330560000000\) \([2]\) \(18874368\) \(3.1578\)  
187200.hh2 187200ed3 \([0, 0, 0, -12168300, 16324882000]\) \(66730743078481/60937500\) \(181958400000000000000\) \([2]\) \(9437184\) \(2.8112\)  
187200.hh3 187200ed4 \([0, 0, 0, -8136300, -8842862000]\) \(19948814692561/231344100\) \(690789781094400000000\) \([2, 2]\) \(9437184\) \(2.8112\)  
187200.hh4 187200ed5 \([0, 0, 0, -1656300, -22541582000]\) \(-168288035761/73415764890\) \(-219218299309301760000000\) \([2]\) \(18874368\) \(3.1578\)  
187200.hh5 187200ed2 \([0, 0, 0, -936300, 128338000]\) \(30400540561/15210000\) \(45416816640000000000\) \([2, 2]\) \(4718592\) \(2.4647\)  
187200.hh6 187200ed1 \([0, 0, 0, 215700, 15442000]\) \(371694959/249600\) \(-745301606400000000\) \([2]\) \(2359296\) \(2.1181\) \(\Gamma_0(N)\)-optimal