# Properties

 Label 10073.a.10073.1 Conductor $10073$ Discriminant $-10073$ Mordell-Weil group $$\Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + (x^2 + x + 1)y = -x^5 + 2x^4 - 3x^2$ (homogenize, simplify) $y^2 + (x^2z + xz^2 + z^3)y = -x^5z + 2x^4z^2 - 3x^2z^4$ (dehomogenize, simplify) $y^2 = -4x^5 + 9x^4 + 2x^3 - 9x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, 0, 2, -1]), R([1, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, 0, 2, -1], R![1, 1, 1]);

sage: X = HyperellipticCurve(R([1, 2, -9, 2, 9, -4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10073$$ $$=$$ $$7 \cdot 1439$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-10073$$ $$=$$ $$- 7 \cdot 1439$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$500$$ $$=$$ $$2^{2} \cdot 5^{3}$$ $$I_4$$ $$=$$ $$7801$$ $$=$$ $$29 \cdot 269$$ $$I_6$$ $$=$$ $$1091757$$ $$=$$ $$3 \cdot 17 \cdot 21407$$ $$I_{10}$$ $$=$$ $$-1289344$$ $$=$$ $$- 2^{7} \cdot 7 \cdot 1439$$ $$J_2$$ $$=$$ $$125$$ $$=$$ $$5^{3}$$ $$J_4$$ $$=$$ $$326$$ $$=$$ $$2 \cdot 163$$ $$J_6$$ $$=$$ $$644$$ $$=$$ $$2^{2} \cdot 7 \cdot 23$$ $$J_8$$ $$=$$ $$-6444$$ $$=$$ $$- 2^{2} \cdot 3^{2} \cdot 179$$ $$J_{10}$$ $$=$$ $$-10073$$ $$=$$ $$- 7 \cdot 1439$$ $$g_1$$ $$=$$ $$-30517578125/10073$$ $$g_2$$ $$=$$ $$-636718750/10073$$ $$g_3$$ $$=$$ $$-1437500/1439$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ 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
$$(1 : 0 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$ $$(1 : -1 : 1)$$
$$(1 : -2 : 1)$$ $$(2 : -3 : 1)$$ $$(2 : -4 : 1)$$ $$(1 : -10 : 4)$$ $$(1 : -74 : 4)$$
Known points
$$(1 : 0 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : -1 : 1)$$ $$(1 : -1 : 1)$$
$$(1 : -2 : 1)$$ $$(2 : -3 : 1)$$ $$(2 : -4 : 1)$$ $$(1 : -10 : 4)$$ $$(1 : -74 : 4)$$
Known points
$$(1 : 0 : 0)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(-1 : -1 : 1)$$ $$(-1 : 1 : 1)$$ $$(1 : -1 : 1)$$
$$(1 : 1 : 1)$$ $$(2 : -1 : 1)$$ $$(2 : 1 : 1)$$ $$(1 : -64 : 4)$$ $$(1 : 64 : 4)$$

magma: [C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-74,4],C![1,-10,4],C![1,-2,1],C![1,-1,1],C![1,0,0],C![2,-4,1],C![2,-3,1]]; // minimal model

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-64,4],C![1,64,4],C![1,-1,1],C![1,1,1],C![1,0,0],C![2,-1,1],C![2,1,1]]; // simplified model

Number of rational Weierstrass points: $$1$$

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
$$(0 : 0 : 1) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2$$ $$0.163604$$ $$\infty$$
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.132689$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 0 : 1) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2$$ $$0.163604$$ $$\infty$$
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2z^3$$ $$0.132689$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z - 3xz^2 + z^3$$ $$0.163604$$ $$\infty$$
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^2z + xz^2 - 3z^3$$ $$0.132689$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.017404$$ Real period: $$24.24596$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.421984$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 7 T^{2} )$$
$$1439$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 36 T + 1439 T^{2} )$$

## 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 $$\Q$$.