# Properties

 Label 726.a.1452.1 Conductor $726$ Discriminant $-1452$ Mordell-Weil group $$\Z/{10}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + 1)y = 2x^5 + 2x^4 + 6x^3 - 2x^2 - x$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = 2x^5z + 2x^4z^2 + 6x^3z^3 - 2x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = 8x^5 + 9x^4 + 24x^3 - 6x^2 - 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 6, 2, 2]), R([1, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -2, 6, 2, 2], R![1, 0, 1]);

sage: X = HyperellipticCurve(R([1, -4, -6, 24, 9, 8]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$726$$ $$=$$ $$2 \cdot 3 \cdot 11^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-1452$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 11^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$760$$ $$=$$ $$2^{3} \cdot 5 \cdot 19$$ $$I_4$$ $$=$$ $$-69236$$ $$=$$ $$- 2^{2} \cdot 19 \cdot 911$$ $$I_6$$ $$=$$ $$-16142609$$ $$=$$ $$- 7^{3} \cdot 19 \cdot 2477$$ $$I_{10}$$ $$=$$ $$-5808$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 11^{2}$$ $$J_2$$ $$=$$ $$380$$ $$=$$ $$2^{2} \cdot 5 \cdot 19$$ $$J_4$$ $$=$$ $$17556$$ $$=$$ $$2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 19$$ $$J_6$$ $$=$$ $$702601$$ $$=$$ $$19 \cdot 36979$$ $$J_8$$ $$=$$ $$-10306189$$ $$=$$ $$- 19^{2} \cdot 28549$$ $$J_{10}$$ $$=$$ $$-1452$$ $$=$$ $$- 2^{2} \cdot 3 \cdot 11^{2}$$ $$g_1$$ $$=$$ $$-1980879200000/363$$ $$g_2$$ $$=$$ $$-7297976000/11$$ $$g_3$$ $$=$$ $$-25363896100/363$$

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

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

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

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

Number of rational Weierstrass points: $$1$$

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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - z^3$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$15.12408$$ Tamagawa product: $$2$$ Torsion order: $$10$$ Leading coefficient: $$0.302481$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

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

## 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

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 66.c
Elliptic curve isogeny class 11.a

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$5$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

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