LMFDB - Genus 2 curve 40000.e.200000.1

Show commands: Magma / SageMath

Minimal equation

Simplified equation

$y^2 + x^3y = x^5 - 5x^3 - 10x^2 - 8x - 2$	(homogenize, simplify)
$y^2 + x^3y = x^5z - 5x^3z^3 - 10x^2z^4 - 8xz^5 - 2z^6$	(dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 20x^3 - 40x^2 - 32x - 8$	(homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -8, -10, -5, 0, 1]), R([0, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -8, -10, -5, 0, 1], R![0, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-8, -32, -40, -20, 0, 4, 1]))

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

Invariants

Conductor:	$ N $	$=$	$40000$	$=$	$ 2^{6} \cdot 5^{4} $	magma: Conductor(LSeries(C)); Factorization($1);
Discriminant:	$ \Delta $	$=$	$-200000$	$=$	$ - 2^{6} \cdot 5^{5} $	magma: Discriminant(C); Factorization(Integers()!$1);

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

$ I_2 $	$=$	$20$	$=$	$ 2^{2} \cdot 5 $
$ I_4 $	$=$	$-20$	$=$	$ - 2^{2} \cdot 5 $
$ I_6 $	$=$	$-40$	$=$	$ - 2^{3} \cdot 5 $
$ I_{10} $	$=$	$8$	$=$	$ 2^{3} $
$ J_2 $	$=$	$100$	$=$	$ 2^{2} \cdot 5^{2} $
$ J_4 $	$=$	$750$	$=$	$ 2 \cdot 3 \cdot 5^{3} $
$ J_6 $	$=$	$-2500$	$=$	$ - 2^{2} \cdot 5^{4} $
$ J_8 $	$=$	$-203125$	$=$	$ - 5^{6} \cdot 13 $
$ J_{10} $	$=$	$200000$	$=$	$ 2^{6} \cdot 5^{5} $
$ g_1 $	$=$	$50000$
$ g_2 $	$=$	$3750$
$ g_3 $	$=$	$-125$

(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

$\mathrm{Aut}(X)$	$\simeq$	$C_4$	magma: AutomorphismGroup(C); IdentifyGroup($1);
$\mathrm{Aut}(X_{\overline{\Q}})$	$\simeq$	$GL(2,3)$	magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Rational points

Known points
$(1 : 0 : 0)$	$(1 : -1 : 0)$	$(-1 : 0 : 1)$	$(-1 : 1 : 1)$	$(-1 : -2 : 2)$	$(-1 : 3 : 2)$
$(-3 : 11 : 1)$	$(-3 : 16 : 1)$

Known points
$(1 : 0 : 0)$	$(1 : -1 : 0)$	$(-1 : 0 : 1)$	$(-1 : 1 : 1)$	$(-1 : -2 : 2)$	$(-1 : 3 : 2)$
$(-3 : 11 : 1)$	$(-3 : 16 : 1)$

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

magma: [C![-3,11,1],C![-3,16,1],C![-1,-2,2],C![-1,0,1],C![-1,1,1],C![-1,3,2],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-3,-5,1],C![-3,5,1],C![-1,-5,2],C![-1,-1,1],C![-1,1,1],C![-1,5,2],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: $\Z \oplus \Z \oplus \Z/{2}\Z$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator	$D_0$						Height	Order
$(-1 : 0 : 1) - (1 : -1 : 0)$	$z (x + z)$	$=$	$0,$	$y$	$=$	$0$	$0.355997$	$\infty$
$(-1 : 0 : 1) - (1 : 0 : 0)$	$z (x + z)$	$=$	$0,$	$y$	$=$	$-x^3 - z^3$	$0.355997$	$\infty$
$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$	$x^2 + 2xz + 2z^2$	$=$	$0,$	$y$	$=$	$-xz^2 - 2z^3$	$0$	$2$

Generator	$D_0$						Height	Order
$(-1 : 0 : 1) - (1 : -1 : 0)$	$z (x + z)$	$=$	$0,$	$y$	$=$	$0$	$0.355997$	$\infty$
$(-1 : 0 : 1) - (1 : 0 : 0)$	$z (x + z)$	$=$	$0,$	$y$	$=$	$-x^3 - z^3$	$0.355997$	$\infty$
$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$	$x^2 + 2xz + 2z^2$	$=$	$0,$	$y$	$=$	$-xz^2 - 2z^3$	$0$	$2$

Generator	$D_0$						Height	Order
$(-1 : -1 : 1) - (1 : -1 : 0)$	$z (x + z)$	$=$	$0,$	$y$	$=$	$x^3$	$0.355997$	$\infty$
$(-1 : -1 : 1) - (1 : 1 : 0)$	$z (x + z)$	$=$	$0,$	$y$	$=$	$-x^3 - 2z^3$	$0.355997$	$\infty$
$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$	$x^2 + 2xz + 2z^2$	$=$	$0,$	$y$	$=$	$x^3 - 2xz^2 - 4z^3$	$0$	$2$

2-torsion field: $\Q(\zeta_{20})$

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	$2$
Mordell-Weil rank:	$2$
2-Selmer rank:	$3$
Regulator:	$ 0.126734 $
Real period:	$ 11.35026 $
Tamagawa product:	$ 2 $
Torsion order:	$ 2 $
Leading coefficient:	$ 0.719234 $
Analytic order of Ш:	$ 1 $ (rounded)
Order of Ш:	square

Local invariants

Prime	ord($N$)	ord($\Delta$)	Tamagawa	L-factor	Cluster picture
$2$	$6$	$6$	$1$	$1 + 2 T + 2 T^{2}$
$5$	$4$	$5$	$2$	$1$

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?
$2$	2.90.7	yes
$3$	3.3240.15	no

Sato-Tate group

$\mathrm{ST}$	$\simeq$	$J(C_4)$
$\mathrm{ST}^0$	$\simeq$	$\mathrm{U}(1)$

Decomposition of the Jacobian

Splits over the number field $\Q (b) \simeq $ 4.4.8000.1 with defining polynomial:
$x^{4} - 10 x^{2} + 20$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
Elliptic curve isogeny class 4.4.8000.1-25.1-b

magma: HeuristicDecompositionFactors(C);

Endomorphisms of the Jacobian

Of $\GL_2$-type over $\Q$

Endomorphism ring over $\Q$:

$\End (J_{})$	$\simeq$	$\Z [\sqrt{-1}]$
$\End (J_{}) \otimes \Q $	$\simeq$	$\Q(\sqrt{-1}) $
$\End (J_{}) \otimes \R$	$\simeq$	$\C$

Smallest field over which all endomorphisms are defined:
Galois number field $K = \Q (a) \simeq $ 8.0.64000000.1 with defining polynomial $x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16$

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

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

$\End (J_{\overline{\Q}})$	$\simeq$	a maximal order of $\End (J_{\overline{\Q}}) \otimes \Q$
$\End (J_{\overline{\Q}}) \otimes \Q $	$\simeq$	$\mathrm{M}_2($$\Q(\sqrt{-2}) $$)$
$\End (J_{\overline{\Q}}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\C)$

Remainder of the endomorphism lattice by field

Over subfield $F \simeq $ $\Q(\sqrt{-2}) $ with generator $-\frac{1}{4} a^{5}$ with minimal polynomial $x^{2} + 2$:

$\End (J_{F})$	$\simeq$	the maximal order of $\End (J_{F}) \otimes \Q$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\zeta_{8})$ (CM)
$\End (J_{F}) \otimes \R$	$\simeq$	$\C \times \C$

Sato Tate group: $C_4$
Not of $\GL_2$-type, simple

Over subfield $F \simeq $ $\Q(\sqrt{-10}) $ with generator $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + a^{3}$ with minimal polynomial $x^{2} + 10$:

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

Sato Tate group: $C_{4,1}$
Of $\GL_2$-type, simple

Over subfield $F \simeq $ $\Q(\sqrt{5}) $ with generator $\frac{1}{8} a^{6} - \frac{1}{4} a^{4}$ with minimal polynomial $x^{2} - x - 1$:

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

Sato Tate group: $J(C_2)$
Of $\GL_2$-type, simple

Over subfield $F \simeq $ $\Q(\sqrt{-2}, \sqrt{5})$ with generator $-\frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$ with minimal polynomial $x^{4} + 6 x^{2} + 4$:

$\End (J_{F})$	$\simeq$	the maximal order of $\End (J_{F}) \otimes \Q$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\Q(\zeta_{8})$ (CM)
$\End (J_{F}) \otimes \R$	$\simeq$	$\C \times \C$

Sato Tate group: $C_2$
Not of $\GL_2$-type, simple

Over subfield $F \simeq $ 4.4.8000.1 with generator $\frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{2} a^{3} - 2 a$ with minimal polynomial $x^{4} - 10 x^{2} + 20$:

$\End (J_{F})$	$\simeq$	a non-Eichler order of index $4$ in a maximal order of $\End (J_{F}) \otimes \Q$
$\End (J_{F}) \otimes \Q $	$\simeq$	$\mathrm{M}_2($$\Q$$)$
$\End (J_{F}) \otimes \R$	$\simeq$	$\mathrm{M}_2 (\R)$

Sato Tate group: $C_{2,1}$
Not of $\GL_2$-type, not simple

Over subfield $F \simeq $ $\Q(\zeta_{5})$ with generator $\frac{1}{2} a^{2}$ with minimal polynomial $x^{4} - x^{3} + x^{2} - x + 1$:

$\End (J_{F})$	$\simeq$	a maximal order of $\End (J_{F}) \otimes \Q$
$\End (J_{F}) \otimes \Q $	$\simeq$	the quaternion algebra over $\Q$ of discriminant 2
$\End (J_{F}) \otimes \R$	$\simeq$	$\H$

Sato Tate group: $J(C_1)$
Not 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 $\F_{q}$

Conductor:	\( N \)	\(=\)	\(40000\)	\(=\)	\( 2^{6} \cdot 5^{4} \)	magma: Conductor(LSeries(C)); Factorization($1);
Discriminant:	\( \Delta \)	\(=\)	\(-200000\)	\(=\)	\( - 2^{6} \cdot 5^{5} \)	magma: Discriminant(C); Factorization(Integers()!$1);

\( I_2 \)	\(=\)	\(20\)	\(=\)	\( 2^{2} \cdot 5 \)
\( I_4 \)	\(=\)	\(-20\)	\(=\)	\( - 2^{2} \cdot 5 \)
\( I_6 \)	\(=\)	\(-40\)	\(=\)	\( - 2^{3} \cdot 5 \)
\( I_{10} \)	\(=\)	\(8\)	\(=\)	\( 2^{3} \)
\( J_2 \)	\(=\)	\(100\)	\(=\)	\( 2^{2} \cdot 5^{2} \)
\( J_4 \)	\(=\)	\(750\)	\(=\)	\( 2 \cdot 3 \cdot 5^{3} \)
\( J_6 \)	\(=\)	\(-2500\)	\(=\)	\( - 2^{2} \cdot 5^{4} \)
\( J_8 \)	\(=\)	\(-203125\)	\(=\)	\( - 5^{6} \cdot 13 \)
\( J_{10} \)	\(=\)	\(200000\)	\(=\)	\( 2^{6} \cdot 5^{5} \)
\( g_1 \)	\(=\)	\(50000\)
\( g_2 \)	\(=\)	\(3750\)
\( g_3 \)	\(=\)	\(-125\)

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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	40000.e.200000.1

Conductor	$40000$
Discriminant	$-200000$
Mordell-Weil group	\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Sato-Tate group	$J(C_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\)	\(\mathrm{M}_2(\C)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\)	\(\mathrm{M}_2(\mathsf{CM})\)
\(\End(J) \otimes \Q\)	\(\mathsf{CM}\)
\(\overline{\Q}\)-simple	no
\(\mathrm{GL}_2\)-type	yes

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_4$	magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$GL(2,3)$	magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((-1 : 1 : 1)\)	\((-1 : -2 : 2)\)	\((-1 : 3 : 2)\)
\((-3 : 11 : 1)\)	\((-3 : 16 : 1)\)

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

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.355997\)	\(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.355997\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + 2xz + 2z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 - 2z^3\)	\(0\)	\(2\)

Generator	$D_0$						Height	Order
\((-1 : -1 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3\)	\(0.355997\)	\(\infty\)
\((-1 : -1 : 1) - (1 : 1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 2z^3\)	\(0.355997\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x^2 + 2xz + 2z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - 2xz^2 - 4z^3\)	\(0\)	\(2\)

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(6\)	\(6\)	\(1\)	\(1 + 2 T + 2 T^{2}\)
\(5\)	\(4\)	\(5\)	\(2\)	\(1\)

\(\mathrm{ST}\)	\(\simeq\)	$J(C_4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{U}(1)\)

\(\End (J_{})\)	\(\simeq\)	\(\Z [\sqrt{-1}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q(\sqrt{-2}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\C)\)

\(\End (J_{F})\)	\(\simeq\)	the maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\zeta_{8})\) (CM)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C \times \C\)

\(\End (J_{F})\)	\(\simeq\)	a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

\(\End (J_{F})\)	\(\simeq\)	a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	the quaternion algebra over \(\Q\) of discriminant 2
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\H\)