# Properties

 Label 578.a.2312.1 Conductor $578$ Discriminant $2312$ Mordell-Weil group $$\Z/{12}\Z$$ 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 for: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x)y = x^5 - 2x^4 + 2x^3 - 2x^2 + x$ (homogenize, simplify) $y^2 + (x^2z + xz^2)y = x^5z - 2x^4z^2 + 2x^3z^3 - 2x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 7x^4 + 10x^3 - 7x^2 + 4x$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([0, 4, -7, 10, -7, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$578$$ $$=$$ $$2 \cdot 17^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$2312$$ $$=$$ $$2^{3} \cdot 17^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$228$$ $$=$$ $$2^{2} \cdot 3 \cdot 19$$ $$I_4$$ $$=$$ $$705$$ $$=$$ $$3 \cdot 5 \cdot 47$$ $$I_6$$ $$=$$ $$135777$$ $$=$$ $$3 \cdot 45259$$ $$I_{10}$$ $$=$$ $$295936$$ $$=$$ $$2^{10} \cdot 17^{2}$$ $$J_2$$ $$=$$ $$57$$ $$=$$ $$3 \cdot 19$$ $$J_4$$ $$=$$ $$106$$ $$=$$ $$2 \cdot 53$$ $$J_6$$ $$=$$ $$-992$$ $$=$$ $$- 2^{5} \cdot 31$$ $$J_8$$ $$=$$ $$-16945$$ $$=$$ $$- 5 \cdot 3389$$ $$J_{10}$$ $$=$$ $$2312$$ $$=$$ $$2^{3} \cdot 17^{2}$$ $$g_1$$ $$=$$ $$601692057/2312$$ $$g_2$$ $$=$$ $$9815229/1156$$ $$g_3$$ $$=$$ $$-402876/289$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

 All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)$$ All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -2 : 1),\, (1 : 2 : 1)$$

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

magma: [C![0,0,1],C![1,-2,1],C![1,0,0],C![1,2,1]]; // simplified model

Number of rational Weierstrass points: $$2$$

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: $$\Z/{12}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$(0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$12$$
Generator $D_0$ Height Order
$$(0 : 0 : 1) + (1 : 2 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z + xz^2$$ $$0$$ $$12$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$13.91029$$ Tamagawa product: $$3$$ Torsion order: $$12$$ Leading coefficient: $$0.289797$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$3$$ $$3$$ $$( 1 - T )( 1 + T + 2 T^{2} )$$
$$17$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## 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 34.a
Elliptic curve isogeny class 17.a

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ 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 $$\Q$$.