# Properties

 Label 1403.a.1403.1 Conductor 1403 Discriminant -1403 Mordell-Weil group $$\Z$$ 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

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

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

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

magma: X,pi:= SimplifiedModel(C);

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

## Invariants

 Conductor: $$N$$ = $$1403$$ = $$23 \cdot 61$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-1403$$ = $$- 23 \cdot 61$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$352$$ = $$2^{5} \cdot 11$$ $$I_4$$ = $$-512$$ = $$- 2^{9}$$ $$I_6$$ = $$-474624$$ = $$- 2^{9} \cdot 3^{2} \cdot 103$$ $$I_{10}$$ = $$-5746688$$ = $$- 2^{12} \cdot 23 \cdot 61$$ $$J_2$$ = $$44$$ = $$2^{2} \cdot 11$$ $$J_4$$ = $$86$$ = $$2 \cdot 43$$ $$J_6$$ = $$956$$ = $$2^{2} \cdot 239$$ $$J_8$$ = $$8667$$ = $$3^{4} \cdot 107$$ $$J_{10}$$ = $$-1403$$ = $$- 23 \cdot 61$$ $$g_1$$ = $$-164916224/1403$$ $$g_2$$ = $$-7325824/1403$$ $$g_3$$ = $$-1850816/1403$$

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

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (-1 : -1 : 1),\, (1 : -1 : 1)$$

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

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.008805$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.008805$$ Real period: $$25.03635$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.220455$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 23 T^{2} )$$
$$61$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 6 T + 61 T^{2} )$$

## 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 $$\Q$$.