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Properties

Label 92442x
Number of curves $4$
Conductor $92442$
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([1, 0, 0, -641724110, 6256980453636]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -641724110, 6256980453636]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -641724110, 6256980453636]) 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 92442x have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(7\)\(1 + T\)
\(31\)\(1 + T\)
\(71\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 + 13 T^{2}\) 1.13.a
\(17\) \( 1 - 7 T + 17 T^{2}\) 1.17.ah
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 7 T + 29 T^{2}\) 1.29.ah
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 92442x do not have complex multiplication.

Modular form 92442.2.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);
 
\(q + q^{2} + q^{3} + q^{4} - 4 q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - 4 q^{10} + 2 q^{11} + q^{12} - 6 q^{13} + q^{14} - 4 q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 92442x

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
92442.n4 92442x1 \([1, 0, 0, -641724110, 6256980453636]\) \(29225959703644585098026815619041/166525600515144466563072\) \(166525600515144466563072\) \([10]\) \(53760000\) \(3.6470\) \(\Gamma_0(N)\)-optimal
92442.n3 92442x2 \([1, 0, 0, -653356750, 6018360109316]\) \(30844295769861594410348211372001/2202100653928823895428923392\) \(2202100653928823895428923392\) \([10]\) \(107520000\) \(3.9936\)  
92442.n2 92442x3 \([1, 0, 0, -4848854030, -126568786842684]\) \(12607850284985940451480268863665121/375761672147788784506617095232\) \(375761672147788784506617095232\) \([2]\) \(268800000\) \(4.4517\)  
92442.n1 92442x4 \([1, 0, 0, -77018028070, -8226909050266324]\) \(50524368854438581121704407622653206881/46954775434734002946337674552\) \(46954775434734002946337674552\) \([2]\) \(537600000\) \(4.7983\)