# Properties

 Label 100121.b.700847.1 Conductor 100121 Discriminant 700847 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

# Related objects

Show commands for: Magma / SageMath

## Simplified equation

 $y^2 + (x^3 + x + 1)y = -x^4 + 3x^2 + 6x + 4$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + 3x^2z^4 + 6xz^5 + 4z^6$ (dehomogenize, simplify) $y^2 = x^6 - 2x^4 + 2x^3 + 13x^2 + 26x + 17$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 6, 3, 0, -1], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 6, 3, 0, -1]), R([1, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([17, 26, 13, 2, -2, 0, 1]))

## Invariants

 Conductor: $$N$$ = $$100121$$ = $$7 \cdot 14303$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$700847$$ = $$7^{2} \cdot 14303$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-3640$$ = $$- 2^{3} \cdot 5 \cdot 7 \cdot 13$$ $$I_4$$ = $$284836$$ = $$2^{2} \cdot 71209$$ $$I_6$$ = $$-342400472$$ = $$- 2^{3} \cdot 42800059$$ $$I_{10}$$ = $$2870669312$$ = $$2^{12} \cdot 7^{2} \cdot 14303$$ $$J_2$$ = $$-455$$ = $$- 5 \cdot 7 \cdot 13$$ $$J_4$$ = $$5659$$ = $$5659$$ $$J_6$$ = $$1397$$ = $$11 \cdot 127$$ $$J_8$$ = $$-8164979$$ = $$- 29 \cdot 281551$$ $$J_{10}$$ = $$700847$$ = $$7^{2} \cdot 14303$$ $$g_1$$ = $$-397979684375/14303$$ $$g_2$$ = $$-10878720125/14303$$ $$g_3$$ = $$5902325/14303$$

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

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(-1 : 1 : 1)$$ $$(2 : 1 : 1)$$ $$(-1 : 9 : 2)$$
$$(-1 : -12 : 2)$$ $$(2 : -12 : 1)$$ $$(-4 : 36 : 3)$$ $$(-4 : 37 : 3)$$

magma: [C![-4,36,3],C![-4,37,3],C![-1,-12,2],C![-1,0,1],C![-1,1,1],C![-1,9,2],C![1,-1,0],C![1,0,0],C![2,-12,1],C![2,1,1]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 1 : 1) + (-1 : 9 : 2) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$(x + z) (2x + z)$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$xz^2 + 5z^3$$ $$0.487205$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 - 3z^3$$ $$0.069459$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.033091$$ Real period: $$14.43539$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$0.955378$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$7$$ $$2$$ $$1$$ $$2$$ $$( 1 - T )( 1 + 4 T + 7 T^{2} )$$
$$14303$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 170 T + 14303 T^{2} )$$

## 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 $$\Q$$.