Properties

Label 30976.a.495616.1
Conductor $30976$
Discriminant $495616$
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

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

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 8, -5, 7, -2, 1]), R([]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 8, -5, 7, -2, 1], R![]);
 
Copy content sage:X = HyperellipticCurve(R([3, 8, -5, 7, -2, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(30976\) \(=\) \( 2^{8} \cdot 11^{2} \)
Copy content magma:Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(30976,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(495616\) \(=\) \( 2^{12} \cdot 11^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(454\) \(=\)  \( 2 \cdot 227 \)
\( I_4 \)  \(=\) \(6832\) \(=\)  \( 2^{4} \cdot 7 \cdot 61 \)
\( I_6 \)  \(=\) \(1220488\) \(=\)  \( 2^{3} \cdot 41 \cdot 61^{2} \)
\( I_{10} \)  \(=\) \(1936\) \(=\)  \( 2^{4} \cdot 11^{2} \)
\( J_2 \)  \(=\) \(908\) \(=\)  \( 2^{2} \cdot 227 \)
\( J_4 \)  \(=\) \(16134\) \(=\)  \( 2 \cdot 3 \cdot 2689 \)
\( J_6 \)  \(=\) \(-2350972\) \(=\)  \( - 2^{2} \cdot 13 \cdot 29 \cdot 1559 \)
\( J_8 \)  \(=\) \(-598747133\) \(=\)  \( - 19 \cdot 31513007 \)
\( J_{10} \)  \(=\) \(495616\) \(=\)  \( 2^{12} \cdot 11^{2} \)
\( g_1 \)  \(=\) \(602738989907/484\)
\( g_2 \)  \(=\) \(94360368561/3872\)
\( g_3 \)  \(=\) \(-30285809047/7744\)

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

Copy content magma:[C![1,0,0]]; // minimal model
 
Copy content magma:[C![1,0,0]]; // 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: \(\Z/{2}\Z\)

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

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

2-torsion field: 3.1.44.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 5.609547 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.402386 \)
Analytic order of Ш: \( 1 \)   (verified)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(8\) \(12\) \(1\) \(1^*\) \(1\) no
\(11\) \(2\) \(2\) \(1\) \(1\) \(( 1 - T )^{2}\) yes

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.120.1 yes
\(3\) 3.72.2 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

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

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{17}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{17}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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);