# Properties

 Label 523.a.523.2 Conductor 523 Discriminant -523 Mordell-Weil group $$\Z/{2}\Z$$ 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: SageMath / Magma

## Simplified equation

 $y^2 + xy = x^5 - 31x^4 - 110x^3 + 21x^2 - x$ (homogenize, simplify) $y^2 + xz^2y = x^5z - 31x^4z^2 - 110x^3z^3 + 21x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 124x^4 - 440x^3 + 85x^2 - 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 21, -110, -31, 1]), R([0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 21, -110, -31, 1], R![0, 1]);

sage: X = HyperellipticCurve(R([0, -4, 85, -440, -124, 4]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$523$$ $$=$$ $$523$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-523$$ $$=$$ $$-523$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$1329600$$ $$=$$ $$2^{6} \cdot 3 \cdot 5^{2} \cdot 277$$ $$I_4$$ $$=$$ $$161357760$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 5 \cdot 179 \cdot 313$$ $$I_6$$ $$=$$ $$70850845209024$$ $$=$$ $$2^{6} \cdot 3^{4} \cdot 37 \cdot 151 \cdot 479 \cdot 5107$$ $$I_{10}$$ $$=$$ $$-2142208$$ $$=$$ $$- 2^{12} \cdot 523$$ $$J_2$$ $$=$$ $$166200$$ $$=$$ $$2^{3} \cdot 3 \cdot 5^{2} \cdot 277$$ $$J_4$$ $$=$$ $$1149254190$$ $$=$$ $$2 \cdot 3^{3} \cdot 5 \cdot 7 \cdot 41 \cdot 14831$$ $$J_6$$ $$=$$ $$10581558955401$$ $$=$$ $$3 \cdot 7 \cdot 431 \cdot 1169103851$$ $$J_8$$ $$=$$ $$109467476288772525$$ $$=$$ $$3^{2} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 6318469049857$$ $$J_{10}$$ $$=$$ $$-523$$ $$=$$ $$-523$$ $$g_1$$ $$=$$ $$-126810465636208320000000000/523$$ $$g_2$$ $$=$$ $$-5276053055713522320000000/523$$ $$g_3$$ $$=$$ $$-292288477352026798440000/523$$

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

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

magma: [C![0,0,1],C![1,0,0]];

Number of rational Weierstrass points: $$2$$

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: $$\Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$0.992796$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.248199$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$523$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 24 T + 523 T^{2} )$$

## 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 $$\Q$$.