# Properties

 Label 57065.a.285325.1 Conductor 57065 Discriminant -285325 Mordell-Weil group $$\Z \times \Z \times \Z$$ 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

 $y^2 + (x^3 + x + 1)y = x^4 - 4x^3 + x^2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = x^4z^2 - 4x^3z^3 + x^2z^4$ (dehomogenize, simplify) $y^2 = x^6 + 6x^4 - 14x^3 + 5x^2 + 2x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 2, 5, -14, 6, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$57065$$ = $$5 \cdot 101 \cdot 113$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-285325$$ = $$- 5^{2} \cdot 101 \cdot 113$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$456$$ = $$2^{3} \cdot 3 \cdot 19$$ $$I_4$$ = $$114468$$ = $$2^{2} \cdot 3 \cdot 9539$$ $$I_6$$ = $$11869608$$ = $$2^{3} \cdot 3 \cdot 494567$$ $$I_{10}$$ = $$-1168691200$$ = $$- 2^{12} \cdot 5^{2} \cdot 101 \cdot 113$$ $$J_2$$ = $$57$$ = $$3 \cdot 19$$ $$J_4$$ = $$-1057$$ = $$- 7 \cdot 151$$ $$J_6$$ = $$-1299$$ = $$- 3 \cdot 433$$ $$J_8$$ = $$-297823$$ = $$- 17 \cdot 17519$$ $$J_{10}$$ = $$-285325$$ = $$- 5^{2} \cdot 101 \cdot 113$$ $$g_1$$ = $$-601692057/285325$$ $$g_2$$ = $$195749001/285325$$ $$g_3$$ = $$4220451/285325$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : -1 : 1)$$ $$(-1 : -2 : 1)$$
$$(1 : -2 : 1)$$ $$(1 : -1 : 2)$$ $$(-1 : 3 : 1)$$ $$(-2 : -4 : 1)$$ $$(2 : -9 : 3)$$ $$(3 : -11 : 2)$$
$$(1 : -12 : 2)$$ $$(-2 : 13 : 1)$$ $$(-2 : 36 : 3)$$ $$(3 : -36 : 2)$$ $$(-2 : -37 : 3)$$ $$(2 : -44 : 3)$$

magma: [C![-2,-37,3],C![-2,-4,1],C![-2,13,1],C![-2,36,3],C![-1,-2,1],C![-1,3,1],C![0,-1,1],C![0,0,1],C![1,-12,2],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,-1,2],C![1,0,0],C![2,-44,3],C![2,-9,3],C![3,-36,2],C![3,-11,2]];

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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -2 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.400353$$ $$\infty$$
$$(0 : 0 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.498323$$ $$\infty$$
$$(-1 : -2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.097232$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.016342$$ Real period: $$17.23534$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.563352$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$5$$ $$2$$ $$1$$ $$2$$ $$( 1 + T )( 1 + 2 T + 5 T^{2} )$$
$$101$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 8 T + 101 T^{2} )$$
$$113$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 113 T^{2} )$$

## 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 $$\Q$$.