# Properties

 Label 4925.b.4925.1 Conductor 4925 Discriminant 4925 Mordell-Weil group $$\Z \times \Z$$ 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

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

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

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

magma: X,pi:= SimplifiedModel(C);

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

## Invariants

 Conductor: $$N$$ = $$4925$$ = $$5^{2} \cdot 197$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$4925$$ = $$5^{2} \cdot 197$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-864$$ = $$- 2^{5} \cdot 3^{3}$$ $$I_4$$ = $$18240$$ = $$2^{6} \cdot 3 \cdot 5 \cdot 19$$ $$I_6$$ = $$-5529600$$ = $$- 2^{13} \cdot 3^{3} \cdot 5^{2}$$ $$I_{10}$$ = $$20172800$$ = $$2^{12} \cdot 5^{2} \cdot 197$$ $$J_2$$ = $$-108$$ = $$- 2^{2} \cdot 3^{3}$$ $$J_4$$ = $$296$$ = $$2^{3} \cdot 37$$ $$J_6$$ = $$984$$ = $$2^{3} \cdot 3 \cdot 41$$ $$J_8$$ = $$-48472$$ = $$- 2^{3} \cdot 73 \cdot 83$$ $$J_{10}$$ = $$4925$$ = $$5^{2} \cdot 197$$ $$g_1$$ = $$-14693280768/4925$$ $$g_2$$ = $$-372874752/4925$$ $$g_3$$ = $$11477376/4925$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : 0 : 1)$$
$$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$ $$(-3 : 7 : 2)$$ $$(-3 : 20 : 2)$$ $$(-11 : 260 : 6)$$ $$(-11 : 1071 : 6)$$

magma: [C![-11,260,6],C![-11,1071,6],C![-3,7,2],C![-3,20,2],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.179718$$ $$\infty$$
$$D_0 - 2 \cdot(1 : -1 : 0)$$ $$z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.072321$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.012038$$ Real period: $$23.18140$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.279065$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$2$$ $$2$$ $$1$$ $$1 + 3 T + 5 T^{2}$$
$$197$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 18 T + 197 T^{2} )$$

## 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 $$\Q$$.