Properties

Label 1109.b.1109.1
Conductor 1109
Discriminant 1109
Mordell-Weil group \(\Z/{7}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^5 - x^4 - x^3 + x^2 + x$ (homogenize, simplify)
$y^2 + z^3y = x^5z - x^4z^2 - x^3z^3 + x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 4x^4 - 4x^3 + 4x^2 + 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, -1, -1, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, -1, -1, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, 4, 4, -4, -4, 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

\( I_2 \)  \(=\) \(992\) \(=\)  \( 2^{5} \cdot 31 \)
\( I_4 \)  \(=\) \(-512\) \(=\)  \( - 2^{9} \)
\( I_6 \)  \(=\) \(-667136\) \(=\)  \( - 2^{9} \cdot 1303 \)
\( I_{10} \)  \(=\) \(4542464\) \(=\)  \( 2^{12} \cdot 1109 \)
\( J_2 \)  \(=\) \(124\) \(=\)  \( 2^{2} \cdot 31 \)
\( J_4 \)  \(=\) \(646\) \(=\)  \( 2 \cdot 17 \cdot 19 \)
\( J_6 \)  \(=\) \(5388\) \(=\)  \( 2^{2} \cdot 3 \cdot 449 \)
\( J_8 \)  \(=\) \(62699\) \(=\)  \( 7 \cdot 13^{2} \cdot 53 \)
\( J_{10} \)  \(=\) \(1109\) \(=\)  \( 1109 \)
\( g_1 \)  \(=\) \(29316250624/1109\)
\( g_2 \)  \(=\) \(1231679104/1109\)
\( g_3 \)  \(=\) \(82845888/1109\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)

magma: [C![0,-1,1],C![0,0,1],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: \(\Z/{7}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(7\)

2-torsion field: 5.1.17744.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 21.60601 \)
Tamagawa product: \( 1 \)
Torsion order:\( 7 \)
Leading coefficient: \( 0.440939 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(1109\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 24 T + 1109 T^{2} )\)

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 \(\Q\).