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The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(33)$ by the involution $W_3$ (see [MR:1373390]), which has discriminant $3\cdot 11^9$.

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 2, 0, 1], R![1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 2, 0, 1]), R([1, 0, 1]))

$y^2 + (x^2 + 1)y = x^5 + 2x^3 + 4x^2 + 2x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$363$$ = $$3 \cdot 11^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-11979$$ = $$-1 \cdot 3^{2} \cdot 11^{3}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1376$$ = $$2^{5} \cdot 43$$ $$I_4$$ = $$-49088$$ = $$-1 \cdot 2^{6} \cdot 13 \cdot 59$$ $$I_6$$ = $$-33691712$$ = $$-1 \cdot 2^{6} \cdot 19 \cdot 103 \cdot 269$$ $$I_{10}$$ = $$-49065984$$ = $$-1 \cdot 2^{12} \cdot 3^{2} \cdot 11^{3}$$ $$J_2$$ = $$172$$ = $$2^{2} \cdot 43$$ $$J_4$$ = $$1744$$ = $$2^{4} \cdot 109$$ $$J_6$$ = $$45841$$ = $$45841$$ $$J_8$$ = $$1210779$$ = $$3^{2} \cdot 29 \cdot 4639$$ $$J_{10}$$ = $$-11979$$ = $$-1 \cdot 3^{2} \cdot 11^{3}$$ $$g_1$$ = $$-150536645632/11979$$ $$g_2$$ = $$-8874253312/11979$$ $$g_3$$ = $$-1356160144/11979$$
Alternative geometric invariants: G2

## Automorphism group

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

## Rational points

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);

This curve is locally solvable everywhere.

magma: [C![-1,-34,4],C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,0,0]];

All rational points: (-1 : -34 : 4), (-1 : -1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : 0 : 0)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$3$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 2 (p = 3), 2 (p = 11) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z \times \Z/{10}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 11.a3
Elliptic curve 33.a2

### Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ an order of index $$3$$ 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 $$\Q$$.