# Properties

 Label 7165.a.7165.1 Conductor 7165 Discriminant -7165 Mordell-Weil group $$\Z \times \Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 0, -8, 6, 4, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$7165$$ = $$5 \cdot 1433$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-7165$$ = $$- 5 \cdot 1433$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$488$$ = $$2^{3} \cdot 61$$ $$I_4$$ = $$12004$$ = $$2^{2} \cdot 3001$$ $$I_6$$ = $$-2096536$$ = $$- 2^{3} \cdot 13 \cdot 19 \cdot 1061$$ $$I_{10}$$ = $$-29347840$$ = $$- 2^{12} \cdot 5 \cdot 1433$$ $$J_2$$ = $$61$$ = $$61$$ $$J_4$$ = $$30$$ = $$2 \cdot 3 \cdot 5$$ $$J_6$$ = $$6284$$ = $$2^{2} \cdot 1571$$ $$J_8$$ = $$95606$$ = $$2 \cdot 7 \cdot 6829$$ $$J_{10}$$ = $$-7165$$ = $$- 5 \cdot 1433$$ $$g_1$$ = $$-844596301/7165$$ $$g_2$$ = $$-1361886/1433$$ $$g_3$$ = $$-23382764/7165$$

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

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

## 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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-2 : 0 : 1)$$
$$(1 : -2 : 1)$$ $$(1 : -4 : 2)$$ $$(1 : -5 : 2)$$ $$(-2 : 7 : 1)$$ $$(10 : 225 : 7)$$ $$(10 : -1568 : 7)$$

magma: [C![-2,0,1],C![-2,7,1],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![10,-1568,7],C![10,225,7]];

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
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.277008$$ $$\infty$$
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.228372$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0$$ $$2$$

2-torsion field: splitting field of $$x^{6} - x^{5} - 25 x^{4} - 84 x^{3} + 42 x^{2} - 8 x + 24$$ with Galois group $S_4\times C_2$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$3$$ Regulator: $$0.058535$$ Real period: $$23.25245$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.340272$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 5 T^{2} )$$
$$1433$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 34 T + 1433 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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