LMFDB - Genus 2 curve 932261.a.932261.1

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Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -2x^4 + 19x^3 - 71x^2 + 98x - 46$	(homogenize, simplify)
$y^2 + (x^3 + z^3)y = -2x^4z^2 + 19x^3z^3 - 71x^2z^4 + 98xz^5 - 46z^6$	(dehomogenize, simplify)
$y^2 = x^6 - 8x^4 + 78x^3 - 284x^2 + 392x - 183$	(homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-46, 98, -71, 19, -2]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-46, 98, -71, 19, -2], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-183, 392, -284, 78, -8, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

Conductor:	$ N $	$=$	$932261$	$=$	$ 11 \cdot 84751 $	magma: Conductor(LSeries(C)); Factorization($1);
Discriminant:	$ \Delta $	$=$	$-932261$	$=$	$ - 11 \cdot 84751 $	magma: Discriminant(C); Factorization(Integers()!$1);

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

$ I_2 $	$=$	$22036$	$=$	$ 2^{2} \cdot 7 \cdot 787 $
$ I_4 $	$=$	$-129335$	$=$	$ - 5 \cdot 25867 $
$ I_6 $	$=$	$-943515931$	$=$	$ -943515931 $
$ I_{10} $	$=$	$-119329408$	$=$	$ - 2^{7} \cdot 11 \cdot 84751 $
$ J_2 $	$=$	$5509$	$=$	$ 7 \cdot 787 $
$ J_4 $	$=$	$1269934$	$=$	$ 2 \cdot 17 \cdot 41 \cdot 911 $
$ J_6 $	$=$	$391878820$	$=$	$ 2^{2} \cdot 5 \cdot 41 \cdot 53 \cdot 71 \cdot 127 $
$ J_8 $	$=$	$136532013756$	$=$	$ 2^{2} \cdot 3 \cdot 41 \cdot 277504093 $
$ J_{10} $	$=$	$-932261$	$=$	$ - 11 \cdot 84751 $
$ g_1 $	$=$	$-5074156546952986549/932261$
$ g_2 $	$=$	$-212324186037072886/932261$
$ g_3 $	$=$	$-11893162050364420/932261$

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

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

Known points
$(1 : 0 : 0)$	$(1 : -1 : 0)$	$(2 : -2 : 1)$	$(2 : -7 : 1)$	$(3 : -16 : 2)$	$(3 : -19 : 2)$
$(4 : -45 : 3)$	$(4 : -46 : 3)$	$(-122 : 2762 : 3)$	$(-122 : 1813059 : 3)$

Known points
$(1 : 0 : 0)$	$(1 : -1 : 0)$	$(2 : -2 : 1)$	$(2 : -7 : 1)$	$(3 : -16 : 2)$	$(3 : -19 : 2)$
$(4 : -45 : 3)$	$(4 : -46 : 3)$	$(-122 : 2762 : 3)$	$(-122 : 1813059 : 3)$

Known points
$(1 : -1 : 0)$	$(1 : 1 : 0)$	$(3 : -3 : 2)$	$(3 : 3 : 2)$	$(4 : -1 : 3)$	$(4 : 1 : 3)$
$(2 : -5 : 1)$	$(2 : 5 : 1)$	$(-122 : -1810297 : 3)$	$(-122 : 1810297 : 3)$

magma: [C![-122,2762,3],C![-122,1813059,3],C![1,-1,0],C![1,0,0],C![2,-7,1],C![2,-2,1],C![3,-19,2],C![3,-16,2],C![4,-46,3],C![4,-45,3]]; // minimal model

magma: [C![-122,-1810297,3],C![-122,1810297,3],C![1,-1,0],C![1,1,0],C![2,-5,1],C![2,5,1],C![3,-3,2],C![3,3,2],C![4,-1,3],C![4,1,3]]; // simplified model

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: $\Z \oplus \Z \oplus \Z$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator	$D_0$						Height	Order
$(2 : -7 : 1) + (4 : -45 : 3) - (1 : -1 : 0) - (1 : 0 : 0)$	$(x - 2z) (3x - 4z)$	$=$	$0,$	$y$	$=$	$-8xz^2 + 9z^3$	$1.317139$	$\infty$
$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$	$2x^2 + 10xz - 17z^2$	$=$	$0,$	$2y$	$=$	$-51xz^2 + 65z^3$	$1.645093$	$\infty$
$(4 : -46 : 3) - (1 : 0 : 0)$	$z (3x - 4z)$	$=$	$0,$	$3y$	$=$	$-3x^3 + 2z^3$	$0.205701$	$\infty$

Generator	$D_0$						Height	Order
$(2 : -7 : 1) + (4 : -45 : 3) - (1 : -1 : 0) - (1 : 0 : 0)$	$(x - 2z) (3x - 4z)$	$=$	$0,$	$y$	$=$	$-8xz^2 + 9z^3$	$1.317139$	$\infty$
$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$	$2x^2 + 10xz - 17z^2$	$=$	$0,$	$2y$	$=$	$-51xz^2 + 65z^3$	$1.645093$	$\infty$
$(4 : -46 : 3) - (1 : 0 : 0)$	$z (3x - 4z)$	$=$	$0,$	$3y$	$=$	$-3x^3 + 2z^3$	$0.205701$	$\infty$

Generator	$D_0$						Height	Order
$(2 : -5 : 1) + (4 : 1 : 3) - (1 : -1 : 0) - (1 : 1 : 0)$	$(x - 2z) (3x - 4z)$	$=$	$0,$	$y$	$=$	$x^3 - 16xz^2 + 19z^3$	$1.317139$	$\infty$
$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$	$2x^2 + 10xz - 17z^2$	$=$	$0,$	$2y$	$=$	$x^3 - 102xz^2 + 131z^3$	$1.645093$	$\infty$
$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$	$z (3x - 4z)$	$=$	$0,$	$3y$	$=$	$-5x^3 + 5z^3$	$0.205701$	$\infty$

2-torsion field: 6.4.14916176.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	$3$ (upper bound)
Mordell-Weil rank:	$3$
2-Selmer rank:	$3$
Regulator:	$ 0.384040 $
Real period:	$ 6.904498 $
Tamagawa product:	$ 1 $
Torsion order:	$ 1 $
Leading coefficient:	$ 2.651607 $
Analytic order of Ш:	$ 1 $ (rounded)
Order of Ш:	square

Local invariants

Prime	ord($N$)	ord($\Delta$)	Tamagawa	L-factor	Cluster picture
$11$	$1$	$1$	$1$	$( 1 + T )( 1 + 4 T + 11 T^{2} )$
$84751$	$1$	$1$	$1$	$( 1 - T )( 1 - 395 T + 84751 T^{2} )$

Galois representations

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

Sato-Tate group

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

Decomposition of the Jacobian

Simple over $\overline{\Q}$

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 $\Q$.

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

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Label	932261.a.932261.1

Conductor	$932261$
Discriminant	$-932261$
Mordell-Weil group	\(\Z \oplus \Z \oplus \Z\)
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

Conductor:	\( N \)	\(=\)	\(932261\)	\(=\)	\( 11 \cdot 84751 \)	magma: Conductor(LSeries(C)); Factorization($1);
Discriminant:	\( \Delta \)	\(=\)	\(-932261\)	\(=\)	\( - 11 \cdot 84751 \)	magma: Discriminant(C); Factorization(Integers()!$1);

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Minimal equation

Minimal equation

Simplified equation

Invariants

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

Automorphism group

Rational points

Mordell-Weil group of the Jacobian

BSD invariants

Local invariants

Galois representations

Sato-Tate group

Decomposition of the Jacobian

Endomorphisms of the Jacobian

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

\( I_2 \)	\(=\)	\(22036\)	\(=\)	\( 2^{2} \cdot 7 \cdot 787 \)
\( I_4 \)	\(=\)	\(-129335\)	\(=\)	\( - 5 \cdot 25867 \)
\( I_6 \)	\(=\)	\(-943515931\)	\(=\)	\( -943515931 \)
\( I_{10} \)	\(=\)	\(-119329408\)	\(=\)	\( - 2^{7} \cdot 11 \cdot 84751 \)
\( J_2 \)	\(=\)	\(5509\)	\(=\)	\( 7 \cdot 787 \)
\( J_4 \)	\(=\)	\(1269934\)	\(=\)	\( 2 \cdot 17 \cdot 41 \cdot 911 \)
\( J_6 \)	\(=\)	\(391878820\)	\(=\)	\( 2^{2} \cdot 5 \cdot 41 \cdot 53 \cdot 71 \cdot 127 \)
\( J_8 \)	\(=\)	\(136532013756\)	\(=\)	\( 2^{2} \cdot 3 \cdot 41 \cdot 277504093 \)
\( J_{10} \)	\(=\)	\(-932261\)	\(=\)	\( - 11 \cdot 84751 \)
\( g_1 \)	\(=\)	\(-5074156546952986549/932261\)
\( g_2 \)	\(=\)	\(-212324186037072886/932261\)
\( g_3 \)	\(=\)	\(-11893162050364420/932261\)

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

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((2 : -2 : 1)\)	\((2 : -7 : 1)\)	\((3 : -16 : 2)\)	\((3 : -19 : 2)\)
\((4 : -45 : 3)\)	\((4 : -46 : 3)\)	\((-122 : 2762 : 3)\)	\((-122 : 1813059 : 3)\)

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((3 : -3 : 2)\)	\((3 : 3 : 2)\)	\((4 : -1 : 3)\)	\((4 : 1 : 3)\)
\((2 : -5 : 1)\)	\((2 : 5 : 1)\)	\((-122 : -1810297 : 3)\)	\((-122 : 1810297 : 3)\)

Generator	$D_0$						Height	Order
\((2 : -7 : 1) + (4 : -45 : 3) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - 2z) (3x - 4z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-8xz^2 + 9z^3\)	\(1.317139\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(2x^2 + 10xz - 17z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-51xz^2 + 65z^3\)	\(1.645093\)	\(\infty\)
\((4 : -46 : 3) - (1 : 0 : 0)\)	\(z (3x - 4z)\)	\(=\)	\(0,\)	\(3y\)	\(=\)	\(-3x^3 + 2z^3\)	\(0.205701\)	\(\infty\)

Generator	$D_0$						Height	Order
\((2 : -5 : 1) + (4 : 1 : 3) - (1 : -1 : 0) - (1 : 1 : 0)\)	\((x - 2z) (3x - 4z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - 16xz^2 + 19z^3\)	\(1.317139\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(2x^2 + 10xz - 17z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(x^3 - 102xz^2 + 131z^3\)	\(1.645093\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(z (3x - 4z)\)	\(=\)	\(0,\)	\(3y\)	\(=\)	\(-5x^3 + 5z^3\)	\(0.205701\)	\(\infty\)

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(11\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 4 T + 11 T^{2} )\)
\(84751\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 395 T + 84751 T^{2} )\)

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

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)