Properties

Label 1109.a.1109.1
Conductor $1109$
Discriminant $1109$
Mordell-Weil group trivial
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^5 - 6x^4 - 36x^3 - 6x^2 + 63x - 36$ (homogenize, simplify)
$y^2 + z^3y = x^5z - 6x^4z^2 - 36x^3z^3 - 6x^2z^4 + 63xz^5 - 36z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 24x^4 - 144x^3 - 24x^2 + 252x - 143$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-36, 63, -6, -36, -6, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-36, 63, -6, -36, -6, 1], R![1]);
 
sage: X = HyperellipticCurve(R([-143, 252, -24, -144, -24, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1109\) \(=\) \( 1109 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(1109\) \(=\) \( 1109 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(38880\) \(=\)  \( 2^{5} \cdot 3^{5} \cdot 5 \)
\( I_4 \)  \(=\) \(87301728\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 107 \cdot 2833 \)
\( I_6 \)  \(=\) \(855606760992\) \(=\)  \( 2^{5} \cdot 3^{4} \cdot 347 \cdot 951283 \)
\( I_{10} \)  \(=\) \(4436\) \(=\)  \( 2^{2} \cdot 1109 \)
\( J_2 \)  \(=\) \(19440\) \(=\)  \( 2^{4} \cdot 3^{5} \cdot 5 \)
\( J_4 \)  \(=\) \(1196112\) \(=\)  \( 2^{4} \cdot 3 \cdot 24919 \)
\( J_6 \)  \(=\) \(510249312\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 727 \cdot 2437 \)
\( J_8 \)  \(=\) \(2122140677184\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 11^{2} \cdot 30448529 \)
\( J_{10} \)  \(=\) \(1109\) \(=\)  \( 1109 \)
\( g_1 \)  \(=\) \(2776395315422822400000/1109\)
\( g_2 \)  \(=\) \(8787404722987008000/1109\)
\( g_3 \)  \(=\) \(192830154395443200/1109\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0)\)

magma: [C![1,0,0]];
 

Number of rational Weierstrass points: \(1\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

Mordell-Weil group of the Jacobian

Group structure: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 5.1.17744.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 0.288506 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.288506 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(1109\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 50 T + 1109 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).