Properties

Label 177813.a.533439.1
Conductor $177813$
Discriminant $-533439$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
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

$y^2 + (x^2 + x + 1)y = -x^6 + 2x^5 - 5x^4 + 6x^3 - 8x^2 + 4x - 2$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = -x^6 + 2x^5z - 5x^4z^2 + 6x^3z^3 - 8x^2z^4 + 4xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 + 8x^5 - 19x^4 + 26x^3 - 29x^2 + 18x - 7$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 4, -8, 6, -5, 2, -1]), R([1, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 4, -8, 6, -5, 2, -1], R![1, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-7, 18, -29, 26, -19, 8, -4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(177813\) \(=\) \( 3^{2} \cdot 23 \cdot 859 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-533439\) \(=\) \( - 3^{3} \cdot 23 \cdot 859 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2860\) \(=\)  \( 2^{2} \cdot 5 \cdot 11 \cdot 13 \)
\( I_4 \)  \(=\) \(55321\) \(=\)  \( 7^{2} \cdot 1129 \)
\( I_6 \)  \(=\) \(51042499\) \(=\)  \( 37 \cdot 41 \cdot 33647 \)
\( I_{10} \)  \(=\) \(68280192\) \(=\)  \( 2^{7} \cdot 3^{3} \cdot 23 \cdot 859 \)
\( J_2 \)  \(=\) \(715\) \(=\)  \( 5 \cdot 11 \cdot 13 \)
\( J_4 \)  \(=\) \(18996\) \(=\)  \( 2^{2} \cdot 3 \cdot 1583 \)
\( J_6 \)  \(=\) \(595008\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 1033 \)
\( J_8 \)  \(=\) \(16145676\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 149497 \)
\( J_{10} \)  \(=\) \(533439\) \(=\)  \( 3^{3} \cdot 23 \cdot 859 \)
\( g_1 \)  \(=\) \(186865965446875/533439\)
\( g_2 \)  \(=\) \(2314509840500/177813\)
\( g_3 \)  \(=\) \(33798107200/59271\)

  (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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

Copy content magma:[]; // minimal model
 
Copy content magma:[]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\R$ and $\Q_{3}$.

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: \(\Z \oplus \Z/{2}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(1.306707\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(1.306707\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2 - z^3\) \(1.306707\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - xz^2 + z^3\) \(0\) \(2\)

2-torsion field: 6.4.1171017147.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 1.306707 \)
Real period: \( 4.892365 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.598222 \)
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?
\(3\) \(2\) \(3\) \(1\) \(-1\) \(( 1 - T )( 1 + T )\) yes
\(23\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 8 T + 23 T^{2} )\) yes
\(859\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 - 12 T + 859 T^{2} )\) 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?
\(2\) 2.15.1 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);