# Properties

 Label 100234.a.801872.1 Conductor 100234 Discriminant 801872 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

 $y^2 + (x^3 + x)y = x^5 - x^4 - 5x^3 - x^2 + 5x - 3$ (homogenize, simplify) $y^2 + (x^3 + xz^2)y = x^5z - x^4z^2 - 5x^3z^3 - x^2z^4 + 5xz^5 - 3z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 2x^4 - 20x^3 - 3x^2 + 20x - 12$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 5, -1, -5, -1, 1], R![0, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 5, -1, -5, -1, 1]), R([0, 1, 0, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-12, 20, -3, -20, -2, 4, 1]))

## Invariants

 Conductor: $$N$$ = $$100234$$ = $$2 \cdot 23 \cdot 2179$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$801872$$ = $$2^{4} \cdot 23 \cdot 2179$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$8384$$ = $$2^{6} \cdot 131$$ $$I_4$$ = $$-31616$$ = $$- 2^{7} \cdot 13 \cdot 19$$ $$I_6$$ = $$-49935424$$ = $$- 2^{6} \cdot 7 \cdot 11 \cdot 10133$$ $$I_{10}$$ = $$3284467712$$ = $$2^{16} \cdot 23 \cdot 2179$$ $$J_2$$ = $$1048$$ = $$2^{3} \cdot 131$$ $$J_4$$ = $$46092$$ = $$2^{2} \cdot 3 \cdot 23 \cdot 167$$ $$J_6$$ = $$2655225$$ = $$3^{2} \cdot 5^{2} \cdot 11801$$ $$J_8$$ = $$164550834$$ = $$2 \cdot 3^{2} \cdot 7 \cdot 1305959$$ $$J_{10}$$ = $$801872$$ = $$2^{4} \cdot 23 \cdot 2179$$ $$g_1$$ = $$79010794805248/50117$$ $$g_2$$ = $$144165579648/2179$$ $$g_3$$ = $$182265264900/50117$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -3 : 1),\, (-2 : 5 : 1),\, (2 : -7 : 1),\, (-3 : 12 : 1),\, (-3 : 18 : 1)$$

magma: [C![-3,12,1],C![-3,18,1],C![-2,5,1],C![1,-1,0],C![1,0,0],C![2,-7,1],C![2,-3,1]];

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(2 : -7 : 1) - (1 : 0 : 0)$$ $$z (x - 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 + z^3$$ $$1.076362$$ $$\infty$$
$$(-2 : 5 : 1) - (1 : 0 : 0)$$ $$z (x + 2z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - 3z^3$$ $$0.116714$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.125506$$ Real period: $$5.600759$$ Tamagawa product: $$2$$ Torsion order: $$1$$ Leading coefficient: $$1.405863$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$4$$ $$1$$ $$2$$ $$( 1 + T )( 1 + 2 T^{2} )$$
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 6 T + 23 T^{2} )$$
$$2179$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 88 T + 2179 T^{2} )$$

## 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 $$\Q$$.