# Properties

 Label 10115.b.70805.1 Conductor $10115$ Discriminant $70805$ Mordell-Weil group $$\Z/{10}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands: SageMath / Magma

## Simplified equation

 $y^2 + xy = 7x^5 - 12x^4 + 7x^3 - 3x^2 + x$ (homogenize, simplify) $y^2 + xz^2y = 7x^5z - 12x^4z^2 + 7x^3z^3 - 3x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 28x^5 - 48x^4 + 28x^3 - 11x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -3, 7, -12, 7]), R([0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -3, 7, -12, 7], R![0, 1]);

sage: X = HyperellipticCurve(R([0, 4, -11, 28, -48, 28]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$10115$$ $$=$$ $$5 \cdot 7 \cdot 17^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$70805$$ $$=$$ $$5 \cdot 7^{2} \cdot 17^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$184$$ $$=$$ $$2^{3} \cdot 23$$ $$I_4$$ $$=$$ $$-45356$$ $$=$$ $$- 2^{2} \cdot 17 \cdot 23 \cdot 29$$ $$I_6$$ $$=$$ $$-3194113$$ $$=$$ $$- 13 \cdot 17 \cdot 97 \cdot 149$$ $$I_{10}$$ $$=$$ $$283220$$ $$=$$ $$2^{2} \cdot 5 \cdot 7^{2} \cdot 17^{2}$$ $$J_2$$ $$=$$ $$92$$ $$=$$ $$2^{2} \cdot 23$$ $$J_4$$ $$=$$ $$7912$$ $$=$$ $$2^{3} \cdot 23 \cdot 43$$ $$J_6$$ $$=$$ $$163521$$ $$=$$ $$3^{2} \cdot 18169$$ $$J_8$$ $$=$$ $$-11888953$$ $$=$$ $$- 23 \cdot 516911$$ $$J_{10}$$ $$=$$ $$70805$$ $$=$$ $$5 \cdot 7^{2} \cdot 17^{2}$$ $$g_1$$ $$=$$ $$6590815232/70805$$ $$g_2$$ $$=$$ $$6160979456/70805$$ $$g_3$$ $$=$$ $$1384041744/70805$$

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

magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model

magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,1,1]]; // simplified model

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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$10$$
Generator $D_0$ Height Order
$$(1 : 1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$xz^2$$ $$0$$ $$10$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$12.98945$$ Tamagawa product: $$2$$ Torsion order: $$10$$ Leading coefficient: $$0.259789$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 4 T + 5 T^{2} )$$
$$7$$ $$1$$ $$2$$ $$2$$ $$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$17$$ $$2$$ $$2$$ $$1$$ $$1 + 2 T + 17 T^{2}$$

## 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 $$\Q$$.