LMFDB - Genus 2 curve 104976.b.104976.1

Simplified equation

$y^2 + x^3y = 2x^3 + 3$	(homogenize, simplify)
$y^2 + x^3y = 2x^3z^3 + 3z^6$	(dehomogenize, simplify)
$y^2 = x^6 + 8x^3 + 12$	(homogenize, minimize)

sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 0, 0, 2]), R([0, 0, 0, 1]));

magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 0, 0, 2], R![0, 0, 0, 1]);

sage:X = HyperellipticCurve(R([12, 0, 0, 8, 0, 0, 1]))

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

Invariants

Conductor:	$ N $	$=$	$104976$	$=$	$ 2^{4} \cdot 3^{8} $	magma:Conductor(LSeries(C)); Factorization($1);
Discriminant:	$ \Delta $	$=$	$104976$	$=$	$ 2^{4} \cdot 3^{8} $	magma:Discriminant(C); Factorization(Integers()!$1);

G2 invariants

$ I_2 $	$=$	$104$	$=$	$ 2^{3} \cdot 13 $
$ I_4 $	$=$	$837$	$=$	$ 3^{3} \cdot 31 $
$ I_6 $	$=$	$21105$	$=$	$ 3^{2} \cdot 5 \cdot 7 \cdot 67 $
$ I_{10} $	$=$	$-54$	$=$	$ - 2 \cdot 3^{3} $
$ J_2 $	$=$	$312$	$=$	$ 2^{3} \cdot 3 \cdot 13 $
$ J_4 $	$=$	$-966$	$=$	$ - 2 \cdot 3 \cdot 7 \cdot 23 $
$ J_6 $	$=$	$-976$	$=$	$ - 2^{4} \cdot 61 $
$ J_8 $	$=$	$-309417$	$=$	$ - 3 \cdot 17 \cdot 6067 $
$ J_{10} $	$=$	$-104976$	$=$	$ - 2^{4} \cdot 3^{8} $
$ g_1 $	$=$	$-760408064/27$
$ g_2 $	$=$	$22637888/81$
$ g_3 $	$=$	$659776/729$

(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);
$\mathrm{Aut}(X_{\overline{\Q}})$	$\simeq$	$D_6$	magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Rational points

All points: $(1 : 0 : 0),\, (1 : -1 : 0)$

All points: $(1 : -1 : 0),\, (1 : 1 : 0)$

magma:[C![1,-1,0],C![1,0,0]]; // minimal model

magma:[C![1,-1,0],C![1,1,0]]; // 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: $\Z/{3}\Z$

magma:MordellWeilGroupGenus2(Jacobian(C));

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

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

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

2-torsion field: 9.1.24794911296.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	$0$
Mordell-Weil rank:	$0$
2-Selmer rank:	$0$
Regulator:	$ 1 $
Real period:	$ 11.94656 $
Tamagawa product:	$ 1 $
Torsion order:	$ 3 $
Leading coefficient:	$ 1.327396 $
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?
$2$	$4$	$4$	$1$	$-1^*$	$1 + 2 T^{2}$		no
$3$	$8$	$8$	$1$	$-1^*$	$1$		no

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?
$2$	2.40.1	no
$3$	3.2880.4	yes

Sato-Tate group

$\mathrm{ST}$	$\simeq$	$J(E_6)$
$\mathrm{ST}^0$	$\simeq$	$\mathrm{SU}(2)$

Decomposition of the Jacobian

Splits over the number field $\Q (b) \simeq $ 6.2.45349632.2 with defining polynomial:
$x^{6} - 12$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  $y^2 = x^3 - g_4 / 48 x - g_6 / 864$ with
  $g_4 = -144 b^{5} + 540 b^{2}$
  $g_6 = 18144 b^{3} - 53136$
   Conductor norm: 81
  $y^2 = x^3 - g_4 / 48 x - g_6 / 864$ with
  $g_4 = 144 b^{5} + 540 b^{2}$
  $g_6 = -18144 b^{3} - 53136$
   Conductor norm: 81

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

Smallest field over which all endomorphisms are defined:
Galois number field $K = \Q (a)$ with defining polynomial $x^{12} - 3 x^{10} - 30 x^{8} + 77 x^{6} + 510 x^{4} + 537 x^{2} + 169$

Not of $\GL_2$-type over $\overline{\Q}$

Endomorphism ring over $\overline{\Q}$:

$\End (J_{\overline{\Q}})$	$\simeq$	an Eichler order of index $3$ in a maximal order of $\End (J_{\overline{\Q}}) \otimes \Q$
$\End (J_{\overline{\Q}}) \otimes \Q $	$\simeq$	$\mathrm{M}_2($$\Q$$)$
$\End (J_{\overline{\Q}}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\R)$

Remainder of the endomorphism lattice by field

Over subfield $F \simeq $ $\Q(\sqrt{-3}) $ with generator $-\frac{78}{6137} a^{10} + \frac{225}{6137} a^{8} + \frac{2602}{6137} a^{6} - \frac{7122}{6137} a^{4} - \frac{42018}{6137} a^{2} - \frac{20770}{6137}$ with minimal polynomial $x^{2} - x + 1$:

$\End (J_{F})$	$\simeq$	$\Z [\frac{1 + \sqrt{-3}}{2}]$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\sqrt{-3}) $
$\End (J_{F}) \otimes \R$	$\simeq$	$\C$

Sato Tate group: $E_6$
Of $\GL_2$-type, simple

Over subfield $F \simeq $ $\Q(\sqrt{3}) $ with generator $\frac{411}{9386} a^{11} - \frac{1597}{9386} a^{9} - \frac{10653}{9386} a^{7} + \frac{39369}{9386} a^{5} + \frac{172859}{9386} a^{3} + \frac{91617}{9386} a$ with minimal polynomial $x^{2} - 3$:

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

Sato Tate group: $J(E_3)$
Not of $\GL_2$-type, simple

Over subfield $F \simeq $ $\Q(\sqrt{-1}) $ with generator $-\frac{27}{159562} a^{11} - \frac{829}{159562} a^{9} + \frac{5373}{159562} a^{7} + \frac{19449}{159562} a^{5} - \frac{129483}{159562} a^{3} - \frac{283131}{159562} a$ with minimal polynomial $x^{2} + 1$:

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

Sato Tate group: $J(E_3)$
Not of $\GL_2$-type, simple

Over subfield $F \simeq $ 3.1.972.1 with generator $\frac{1}{361} a^{10} - \frac{8}{361} a^{8} + \frac{10}{361} a^{6} + \frac{27}{361} a^{4} + \frac{14}{361} a^{2} + \frac{828}{361}$ with minimal polynomial $x^{3} - 12$:

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

Sato Tate group: $J(E_2)$
Not of $\GL_2$-type, simple

Over subfield $F \simeq $ 3.1.972.1 with generator $-\frac{281}{6137} a^{10} + \frac{1146}{6137} a^{8} + \frac{6823}{6137} a^{6} - \frac{26701}{6137} a^{4} - \frac{109954}{6137} a^{2} - \frac{66361}{6137}$ with minimal polynomial $x^{3} - 12$:

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

Sato Tate group: $J(E_2)$
Not of $\GL_2$-type, simple

Over subfield $F \simeq $ 3.1.972.1 with generator $\frac{264}{6137} a^{10} - \frac{1010}{6137} a^{8} - \frac{6993}{6137} a^{6} + \frac{26242}{6137} a^{4} + \frac{109716}{6137} a^{2} + \frac{52285}{6137}$ with minimal polynomial $x^{3} - 12$:

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

Sato Tate group: $J(E_2)$
Not of $\GL_2$-type, simple

Over subfield $F \simeq $ $\Q(\zeta_{12})$ with generator $\frac{1740}{79781} a^{11} - \frac{13989}{159562} a^{9} - \frac{43932}{79781} a^{7} + \frac{344361}{159562} a^{5} + \frac{702280}{79781} a^{3} + \frac{637179}{159562} a$ with minimal polynomial $x^{4} - x^{2} + 1$:

$\End (J_{F})$	$\simeq$	$\Z [\frac{1 + \sqrt{-3}}{2}]$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\sqrt{-3}) $
$\End (J_{F}) \otimes \R$	$\simeq$	$\C$

Sato Tate group: $E_3$
Of $\GL_2$-type, simple

Over subfield $F \simeq $ 6.0.2834352.1 with generator $-\frac{43}{6137} a^{10} + \frac{99}{12274} a^{8} + \frac{1774}{6137} a^{6} - \frac{5343}{12274} a^{4} - \frac{31686}{6137} a^{2} - \frac{54165}{12274}$ with minimal polynomial $x^{6} + 12$:

$\End (J_{F})$	$\simeq$	$\Z [\frac{1 + \sqrt{-3}}{2}]$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\sqrt{-3}) $
$\End (J_{F}) \otimes \R$	$\simeq$	$\C$

Sato Tate group: $E_2$
Of $\GL_2$-type, simple

Over subfield $F \simeq $ 6.0.15116544.4 with generator $\frac{21}{159562} a^{11} - \frac{1}{722} a^{10} - \frac{844}{79781} a^{9} + \frac{4}{361} a^{8} + \frac{4209}{79781} a^{7} - \frac{5}{361} a^{6} + \frac{11332}{79781} a^{5} - \frac{27}{722} a^{4} - \frac{151231}{159562} a^{3} - \frac{7}{361} a^{2} - \frac{325657}{159562} a - \frac{414}{361}$ with minimal polynomial $x^{6} - 6 x^{3} + 18$:

$\End (J_{F})$	$\simeq$	$\Z [\sqrt{3}]$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\sqrt{3}) $
$\End (J_{F}) \otimes \R$	$\simeq$	$\R \times \R$

Sato Tate group: $J(E_1)$
Of $\GL_2$-type, simple

Over subfield $F \simeq $ 6.2.45349632.2 with generator $-\frac{3507}{159562} a^{11} + \frac{6580}{79781} a^{9} + \frac{93237}{159562} a^{7} - \frac{162456}{79781} a^{5} - \frac{1534043}{159562} a^{3} - \frac{539936}{79781} a$ with minimal polynomial $x^{6} - 12$:

$\End (J_{F})$	$\simeq$	an order of index $2$ in $\Z \times \Z$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q$ $\times$ $\Q$
$\End (J_{F}) \otimes \R$	$\simeq$	$\R \times \R$

Sato Tate group: $J(E_1)$
Of $\GL_2$-type, not simple

Over subfield $F \simeq $ 6.2.45349632.2 with generator $-\frac{2001}{159562} a^{11} + \frac{6679}{159562} a^{9} + \frac{62279}{159562} a^{7} - \frac{204621}{159562} a^{5} - \frac{962985}{159562} a^{3} - \frac{256889}{159562} a$ with minimal polynomial $x^{6} - 12$:

$\End (J_{F})$	$\simeq$	an order of index $2$ in $\Z \times \Z$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q$ $\times$ $\Q$
$\End (J_{F}) \otimes \R$	$\simeq$	$\R \times \R$

Sato Tate group: $J(E_1)$
Of $\GL_2$-type, not simple

Over subfield $F \simeq $ 6.2.45349632.2 with generator $\frac{162}{4693} a^{11} - \frac{1167}{9386} a^{9} - \frac{4574}{4693} a^{7} + \frac{31149}{9386} a^{5} + \frac{73442}{4693} a^{3} + \frac{78633}{9386} a$ with minimal polynomial $x^{6} - 12$:

$\End (J_{F})$	$\simeq$	an order of index $2$ in $\Z \times \Z$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q$ $\times$ $\Q$
$\End (J_{F}) \otimes \R$	$\simeq$	$\R \times \R$

Sato Tate group: $J(E_1)$
Of $\GL_2$-type, not simple

Over subfield $F \simeq $ 6.0.15116544.4 with generator $\frac{3313}{159562} a^{11} + \frac{281}{12274} a^{10} - \frac{10563}{159562} a^{9} - \frac{573}{6137} a^{8} - \frac{50410}{79781} a^{7} - \frac{6823}{12274} a^{6} + \frac{149748}{79781} a^{5} + \frac{26701}{12274} a^{4} + \frac{830216}{79781} a^{3} + \frac{54977}{6137} a^{2} + \frac{1020453}{159562} a + \frac{66361}{12274}$ with minimal polynomial $x^{6} - 6 x^{3} + 18$:

$\End (J_{F})$	$\simeq$	$\Z [\sqrt{3}]$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\sqrt{3}) $
$\End (J_{F}) \otimes \R$	$\simeq$	$\R \times \R$

Sato Tate group: $J(E_1)$
Of $\GL_2$-type, simple

Over subfield $F \simeq $ 6.0.15116544.4 with generator $-\frac{1667}{79781} a^{11} - \frac{132}{6137} a^{10} + \frac{12251}{159562} a^{9} + \frac{505}{6137} a^{8} + \frac{46201}{79781} a^{7} + \frac{6993}{12274} a^{6} - \frac{161080}{79781} a^{5} - \frac{13121}{6137} a^{4} - \frac{1509201}{159562} a^{3} - \frac{54858}{6137} a^{2} - \frac{347398}{79781} a - \frac{52285}{12274}$ with minimal polynomial $x^{6} - 6 x^{3} + 18$:

$\End (J_{F})$	$\simeq$	$\Z [\sqrt{3}]$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\sqrt{3}) $
$\End (J_{F}) \otimes \R$	$\simeq$	$\R \times \R$

Sato Tate group: $J(E_1)$
Of $\GL_2$-type, simple

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 $\F_{q}$
Belyi maps

Conductor:	\( N \)	\(=\)	\(104976\)	\(=\)	\( 2^{4} \cdot 3^{8} \)	magma:Conductor(LSeries(C)); Factorization($1);
Discriminant:	\( \Delta \)	\(=\)	\(104976\)	\(=\)	\( 2^{4} \cdot 3^{8} \)	magma:Discriminant(C); Factorization(Integers()!$1);

\( I_2 \)	\(=\)	\(104\)	\(=\)	\( 2^{3} \cdot 13 \)
\( I_4 \)	\(=\)	\(837\)	\(=\)	\( 3^{3} \cdot 31 \)
\( I_6 \)	\(=\)	\(21105\)	\(=\)	\( 3^{2} \cdot 5 \cdot 7 \cdot 67 \)
\( I_{10} \)	\(=\)	\(-54\)	\(=\)	\( - 2 \cdot 3^{3} \)
\( J_2 \)	\(=\)	\(312\)	\(=\)	\( 2^{3} \cdot 3 \cdot 13 \)
\( J_4 \)	\(=\)	\(-966\)	\(=\)	\( - 2 \cdot 3 \cdot 7 \cdot 23 \)
\( J_6 \)	\(=\)	\(-976\)	\(=\)	\( - 2^{4} \cdot 61 \)
\( J_8 \)	\(=\)	\(-309417\)	\(=\)	\( - 3 \cdot 17 \cdot 6067 \)
\( J_{10} \)	\(=\)	\(-104976\)	\(=\)	\( - 2^{4} \cdot 3^{8} \)
\( g_1 \)	\(=\)	\(-760408064/27\)
\( g_2 \)	\(=\)	\(22637888/81\)
\( g_3 \)	\(=\)	\(659776/729\)

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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

Remainder of the endomorphism lattice by field

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

Label	104976.b.104976.1

Conductor	$104976$
Discriminant	$104976$
Mordell-Weil group	\(\Z/{3}\Z\)
Sato-Tate group	$J(E_6)$
\(\End(J_{\overline{\Q}}) \otimes \R\)	\(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\)	\(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\)	\(\Q\)
\(\overline{\Q}\)-simple	no
\(\mathrm{GL}_2\)-type	no

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$	magma:AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$	magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	Root number*	L-factor	Cluster picture	Tame reduction?
\(2\)	\(4\)	\(4\)	\(1\)	\(-1^*\)	\(1 + 2 T^{2}\)		no
\(3\)	\(8\)	\(8\)	\(1\)	\(-1^*\)	\(1\)		no

\(\mathrm{ST}\)	\(\simeq\)	$J(E_6)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

\(\End (J_{F})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)