Properties

Label 708.a.181248.1
Conductor 708
Discriminant -181248
Mordell-Weil group \(\Z/{2}\Z\)
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

Related objects

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -x^6 - 4x^5 + 9x^4 + 48x^3 - 41x^2 - 98x - 36$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^6 - 4x^5z + 9x^4z^2 + 48x^3z^3 - 41x^2z^4 - 98xz^5 - 36z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 16x^5 + 36x^4 + 194x^3 - 164x^2 - 392x - 143$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-36, -98, -41, 48, 9, -4, -1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-36, -98, -41, 48, 9, -4, -1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-143, -392, -164, 194, 36, -16, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(708\) = \( 2^{2} \cdot 3 \cdot 59 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-181248\) = \( - 2^{10} \cdot 3 \cdot 59 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(468200\) =  \( 2^{3} \cdot 5^{2} \cdot 2341 \)
\( I_4 \)  = \(13875516100\) =  \( 2^{2} \cdot 5^{2} \cdot 138755161 \)
\( I_6 \)  = \(1620683728649416\) =  \( 2^{3} \cdot 1069463 \cdot 189427279 \)
\( I_{10} \)  = \(-742391808\) =  \( - 2^{22} \cdot 3 \cdot 59 \)
\( J_2 \)  = \(58525\) =  \( 5^{2} \cdot 2341 \)
\( J_4 \)  = \(-1820975\) =  \( - 5^{2} \cdot 13^{2} \cdot 431 \)
\( J_6 \)  = \(60952909\) =  \( 109 \cdot 559201 \)
\( J_8 \)  = \(62829762150\) =  \( 2 \cdot 3 \cdot 5^{2} \cdot 47 \cdot 8912023 \)
\( J_{10} \)  = \(-181248\) =  \( - 2^{10} \cdot 3 \cdot 59 \)
\( g_1 \)  = \(-686605237334059580078125/181248\)
\( g_2 \)  = \(365029741228054296875/181248\)
\( g_3 \)  = \(-208774418179643125/181248\)

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

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 2xz - 11z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-15xz^2 + 21z^3\) \(0\) \(2\)

2-torsion field: 6.2.24060672.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 0.325343 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.325343 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(10\) \(2\) \(1\) \(1 + T^{2}\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 3 T^{2} )\)
\(59\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 59 T^{2} )\)

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 \(\Q\).