# Properties

 Label 476.a.952.1 Conductor $476$ Discriminant $-952$ Mordell-Weil group $$\Z/{3}\Z \times \Z/{6}\Z$$ 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

 $y^2 + (x^3 + 1)y = -5x^4 + 7x^3 + 25x^2 - 75x + 54$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -5x^4z^2 + 7x^3z^3 + 25x^2z^4 - 75xz^5 + 54z^6$ (dehomogenize, simplify) $y^2 = x^6 - 20x^4 + 30x^3 + 100x^2 - 300x + 217$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([54, -75, 25, 7, -5]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![54, -75, 25, 7, -5], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([217, -300, 100, 30, -20, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$476$$ $$=$$ $$2^{2} \cdot 7 \cdot 17$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-952$$ $$=$$ $$- 2^{3} \cdot 7 \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$7340$$ $$=$$ $$2^{2} \cdot 5 \cdot 367$$ $$I_4$$ $$=$$ $$1042345$$ $$=$$ $$5 \cdot 208469$$ $$I_6$$ $$=$$ $$2905273355$$ $$=$$ $$5 \cdot 13789 \cdot 42139$$ $$I_{10}$$ $$=$$ $$121856$$ $$=$$ $$2^{10} \cdot 7 \cdot 17$$ $$J_2$$ $$=$$ $$1835$$ $$=$$ $$5 \cdot 367$$ $$J_4$$ $$=$$ $$96870$$ $$=$$ $$2 \cdot 3 \cdot 5 \cdot 3229$$ $$J_6$$ $$=$$ $$-3910340$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 7 \cdot 17 \cdot 31 \cdot 53$$ $$J_8$$ $$=$$ $$-4139817700$$ $$=$$ $$- 2^{2} \cdot 5^{2} \cdot 41398177$$ $$J_{10}$$ $$=$$ $$952$$ $$=$$ $$2^{3} \cdot 7 \cdot 17$$ $$g_1$$ $$=$$ $$20805604708146875/952$$ $$g_2$$ $$=$$ $$299272981175625/476$$ $$g_3$$ $$=$$ $$-27661753375/2$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -4 : 1),\, (2 : -5 : 1)$$ All points: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -4 : 1),\, (2 : -5 : 1)$$ All points: $$(1 : -1 : 0),\, (1 : 1 : 0),\, (2 : -1 : 1),\, (2 : 1 : 1)$$

magma: [C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-4,1]]; // minimal model

magma: [C![1,-1,0],C![1,1,0],C![2,-1,1],C![2,1,1]]; // 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: $$\Z/{3}\Z \times \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 3xz - 12z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$-21xz^2 + 24z^3$$ $$0$$ $$3$$
$$(2 : -4 : 1) - (1 : 0 : 0)$$ $$z (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + 4z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$2x^2 + 3xz - 12z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$-21xz^2 + 24z^3$$ $$0$$ $$3$$
$$(2 : -4 : 1) - (1 : 0 : 0)$$ $$z (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + 4z^3$$ $$0$$ $$6$$
Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$2x^2 + 3xz - 12z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$x^3 - 42xz^2 + 49z^3$$ $$0$$ $$3$$
$$(2 : 1 : 1) - (1 : 1 : 0)$$ $$z (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + 9z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$26.72233$$ Tamagawa product: $$3$$ Torsion order: $$18$$ Leading coefficient: $$0.247429$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$3$$ $$3$$ $$( 1 - T )( 1 + T )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 7 T^{2} )$$
$$17$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 6 T + 17 T^{2} )$$

## 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 14.a
Elliptic curve isogeny class 34.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 $$\Q$$.