# Properties

 Label 27225.a.81675.1 Conductor $27225$ Discriminant $-81675$ Mordell-Weil group $$\Z \times \Z \times \Z/{2}\Z$$ 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

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + 1)y = 2x^4 + 2x^3 + 5x^2 + 3x + 2$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = 2x^4z^2 + 2x^3z^3 + 5x^2z^4 + 3xz^5 + 2z^6$ (dehomogenize, simplify) $y^2 = x^6 + 8x^4 + 10x^3 + 20x^2 + 12x + 9$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 3, 5, 2, 2]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 3, 5, 2, 2], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([9, 12, 20, 10, 8, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$27225$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 11^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-81675$$ $$=$$ $$- 3^{3} \cdot 5^{2} \cdot 11^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$2060$$ $$=$$ $$2^{2} \cdot 5 \cdot 103$$ $$I_4$$ $$=$$ $$56521$$ $$=$$ $$29 \cdot 1949$$ $$I_6$$ $$=$$ $$34621947$$ $$=$$ $$3^{2} \cdot 31^{2} \cdot 4003$$ $$I_{10}$$ $$=$$ $$10454400$$ $$=$$ $$2^{7} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}$$ $$J_2$$ $$=$$ $$515$$ $$=$$ $$5 \cdot 103$$ $$J_4$$ $$=$$ $$8696$$ $$=$$ $$2^{3} \cdot 1087$$ $$J_6$$ $$=$$ $$172224$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 13 \cdot 23$$ $$J_8$$ $$=$$ $$3268736$$ $$=$$ $$2^{7} \cdot 25537$$ $$J_{10}$$ $$=$$ $$81675$$ $$=$$ $$3^{3} \cdot 5^{2} \cdot 11^{2}$$ $$g_1$$ $$=$$ $$1449092592875/3267$$ $$g_2$$ $$=$$ $$47511769960/3267$$ $$g_3$$ $$=$$ $$203013824/363$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 1 : 1)$$ $$(0 : -2 : 1)$$ $$(-1 : -2 : 1)$$ $$(-1 : 2 : 1)$$
$$(7 : 1321 : 8)$$ $$(7 : -2176 : 8)$$
Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 1 : 1)$$ $$(0 : -2 : 1)$$ $$(-1 : -2 : 1)$$ $$(-1 : 2 : 1)$$
$$(7 : 1321 : 8)$$ $$(7 : -2176 : 8)$$
Known points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : -3 : 1)$$ $$(0 : 3 : 1)$$ $$(-1 : -4 : 1)$$ $$(-1 : 4 : 1)$$
$$(7 : -3497 : 8)$$ $$(7 : 3497 : 8)$$

magma: [C![-1,-2,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0],C![7,-2176,8],C![7,1321,8]]; // minimal model

magma: [C![-1,-4,1],C![-1,4,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0],C![7,-3497,8],C![7,3497,8]]; // 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: $$\Z \times \Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 2xz + 3z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$xz^2 - z^3$$ $$0.393627$$ $$\infty$$
$$(0 : -2 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.365989$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 2xz + 3z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$xz^2 - z^3$$ $$0.393627$$ $$\infty$$
$$(0 : -2 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.365989$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + 2xz + 3z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$x^3 + 2xz^2 - z^3$$ $$0.393627$$ $$\infty$$
$$(0 : -3 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 3z^3$$ $$0.365989$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 + xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$3$$ Regulator: $$0.143872$$ Real period: $$9.543118$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$0.686496$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$3$$ $$2$$ $$3$$ $$2$$ $$( 1 + T )^{2}$$
$$5$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$
$$11$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## 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 $$\overline{\Q}$$

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z [\sqrt{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{2})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.