# Properties

 Label 5026.a.35182.1 Conductor 5026 Discriminant 35182 Mordell-Weil group $$\Z \times \Z \times \Z/{2}\Z$$ 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

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Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + (x^3 + 1)y = 4x^5 + 22x^4 + 46x^3 + 28x^2 + 5x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = 4x^5z + 22x^4z^2 + 46x^3z^3 + 28x^2z^4 + 5xz^5$ (dehomogenize, simplify) $y^2 = x^6 + 16x^5 + 88x^4 + 186x^3 + 112x^2 + 20x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 5, 28, 46, 22, 4]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 5, 28, 46, 22, 4], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([1, 20, 112, 186, 88, 16, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$5026$$ $$=$$ $$2 \cdot 7 \cdot 359$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$35182$$ $$=$$ $$2 \cdot 7^{2} \cdot 359$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$62440$$ $$=$$ $$2^{3} \cdot 5 \cdot 7 \cdot 223$$ $$I_4$$ $$=$$ $$1113316$$ $$=$$ $$2^{2} \cdot 278329$$ $$I_6$$ $$=$$ $$22822085992$$ $$=$$ $$2^{3} \cdot 2852760749$$ $$I_{10}$$ $$=$$ $$144105472$$ $$=$$ $$2^{13} \cdot 7^{2} \cdot 359$$ $$J_2$$ $$=$$ $$7805$$ $$=$$ $$5 \cdot 7 \cdot 223$$ $$J_4$$ $$=$$ $$2526654$$ $$=$$ $$2 \cdot 3 \cdot 13 \cdot 29 \cdot 1117$$ $$J_6$$ $$=$$ $$1086135208$$ $$=$$ $$2^{3} \cdot 113 \cdot 491 \cdot 2447$$ $$J_8$$ $$=$$ $$523326215681$$ $$=$$ $$41 \cdot 59 \cdot 2153 \cdot 100483$$ $$J_{10}$$ $$=$$ $$35182$$ $$=$$ $$2 \cdot 7^{2} \cdot 359$$ $$g_1$$ $$=$$ $$591110204777028125/718$$ $$g_2$$ $$=$$ $$12258530733232875/359$$ $$g_3$$ $$=$$ $$675155221982900/359$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(-1 : 0 : 2)$$ $$(-1 : -7 : 2)$$
$$(-4 : 28 : 1)$$ $$(-4 : 35 : 1)$$ $$(-6 : 105 : 1)$$ $$(-6 : 110 : 1)$$

magma: [C![-6,105,1],C![-6,110,1],C![-4,28,1],C![-4,35,1],C![-1,-7,2],C![-1,0,2],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + 5xz - 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-15xz^2 + 3z^3$$ $$0.428002$$ $$\infty$$
$$(-4 : 35 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + 4z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-9xz^2 - z^3$$ $$0.075443$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + 6xz + z^2$$ $$=$$ $$0,$$ $$2y$$ $$=$$ $$-35xz^2 - 7z^3$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$3$$ Regulator: $$0.031070$$ Real period: $$18.25299$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$0.283562$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$7$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + 3 T + 7 T^{2} )$$
$$359$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 12 T + 359 T^{2} )$$

## 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 $$\Q$$.