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Properties

Label 1104.a.17664.1
Conductor $1104$
Discriminant $17664$
Mordell-Weil group \(\Z/{10}\Z\)
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

$y^2 = x^5 - 2x^4 + 4x^3 - 4x^2 + 3x - 1$ (homogenize, simplify)
$y^2 = x^5z - 2x^4z^2 + 4x^3z^3 - 4x^2z^4 + 3xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = x^5 - 2x^4 + 4x^3 - 4x^2 + 3x - 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -4, 4, -2, 1]), R([]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -4, 4, -2, 1], R![]);
 
Copy content sage:X = HyperellipticCurve(R([-1, 3, -4, 4, -2, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1104\) \(=\) \( 2^{4} \cdot 3 \cdot 23 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(17664\) \(=\) \( 2^{8} \cdot 3 \cdot 23 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(88\) \(=\)  \( 2^{3} \cdot 11 \)
\( I_4 \)  \(=\) \(160\) \(=\)  \( 2^{5} \cdot 5 \)
\( I_6 \)  \(=\) \(4888\) \(=\)  \( 2^{3} \cdot 13 \cdot 47 \)
\( I_{10} \)  \(=\) \(69\) \(=\)  \( 3 \cdot 23 \)
\( J_2 \)  \(=\) \(176\) \(=\)  \( 2^{4} \cdot 11 \)
\( J_4 \)  \(=\) \(864\) \(=\)  \( 2^{5} \cdot 3^{3} \)
\( J_6 \)  \(=\) \(-1280\) \(=\)  \( - 2^{8} \cdot 5 \)
\( J_8 \)  \(=\) \(-242944\) \(=\)  \( - 2^{8} \cdot 13 \cdot 73 \)
\( J_{10} \)  \(=\) \(17664\) \(=\)  \( 2^{8} \cdot 3 \cdot 23 \)
\( g_1 \)  \(=\) \(659664896/69\)
\( g_2 \)  \(=\) \(6133248/23\)
\( g_3 \)  \(=\) \(-154880/69\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 1),\, (1 : 1 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 1),\, (1 : 1 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1/2 : 1),\, (1 : 1/2 : 1)\)

Copy content magma:[C![1,-1,1],C![1,0,0],C![1,1,1]]; // minimal model
 
Copy content magma:[C![1,-1/2,1],C![1,0,0],C![1,1/2,1]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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

This curve is locally solvable everywhere.

Copy content 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: \(\Z/{10}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : 1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\((1 : 1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\((1 : 1/2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(1/2z^3\) \(0\) \(10\)

2-torsion field: 6.0.14283.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 8.907497 \)
Tamagawa product: \( 5 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.445374 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(4\) \(8\) \(5\) \(-1^*\) \(1\) yes
\(3\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + T + 3 T^{2} )\) yes
\(23\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 23 T^{2} )\) yes

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.60.1 yes
\(5\) not computed yes

Sato-Tate group

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

Decomposition of the Jacobian

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

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(\Q\).

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);