# Properties

 Label 100261.a.100261.1 Conductor 100261 Discriminant -100261 Mordell-Weil group $$\Z \times \Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

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Show commands for: Magma / SageMath

## Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 7, -1, -16, 1], R![1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 7, -1, -16, 1]), R([1, 1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 7, -1, -16, 1], R![1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -10, 29, -4, -64, 4]))

 $y^2 + (x + 1)y = x^5 - 16x^4 - x^3 + 7x^2 - 3x$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^5z - 16x^4z^2 - x^3z^3 + 7x^2z^4 - 3xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 64x^4 - 4x^3 + 29x^2 - 10x + 1$ (minimize, homogenize)

## Invariants

 $$N$$ = $$100261$$ = $$7 \cdot 14323$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-100261$$ = $$- 7 \cdot 14323$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$28192$$ = $$2^{5} \cdot 881$$ $$I_4$$ = $$-507776$$ = $$- 2^{7} \cdot 3967$$ $$I_6$$ = $$-4691938624$$ = $$- 2^{6} \cdot 37 \cdot 1981393$$ $$I_{10}$$ = $$-410669056$$ = $$- 2^{12} \cdot 7 \cdot 14323$$ $$J_2$$ = $$3524$$ = $$2^{2} \cdot 881$$ $$J_4$$ = $$522730$$ = $$2 \cdot 5 \cdot 13 \cdot 4021$$ $$J_6$$ = $$104271441$$ = $$3 \cdot 907 \cdot 38321$$ $$J_8$$ = $$23551476296$$ = $$2^{3} \cdot 19 \cdot 2017 \cdot 76819$$ $$J_{10}$$ = $$-100261$$ = $$- 7 \cdot 14323$$ $$g_1$$ = $$-543474909254042624/100261$$ $$g_2$$ = $$-22876265307259520/100261$$ $$g_3$$ = $$-1294902814688016/100261$$

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$$C_2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $C_2$

## Rational points

magma: [C![0,-1,1],C![0,0,1],C![1,-42,4],C![1,-38,4],C![1,0,0],C![21,-994,1],C![21,972,1]];

Known points: $$(0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (1 : -38 : 4),\, (1 : -42 : 4),\, (21 : 972 : 1),\, (21 : -994 : 1)$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: $$1$$

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);

This curve is locally solvable everywhere.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z \times \Z$$

Generator Height Order
$$(-4x + z) x$$ $$=$$ $$0,$$ $$8y$$ $$=$$ $$11xz^2 - 8z^3$$ $$1.274501$$ $$\infty$$
$$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.170227$$ $$\infty$$

## BSD invariants

 Analytic rank: $$2$$   (upper bound) Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.215449$$ Real period: $$6.839751$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.473618$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 7 T^{2} )$$
$$14323$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 140 T + 14323 T^{2} )$$

## 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 $$\Q$$.