# Properties

 Label 6845.a.6845.1 Conductor 6845 Discriminant 6845 Mordell-Weil group $$\Z \times \Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 + x^3y = x^5 - 7x^3 - 16x^2 - 15x - 5$ (homogenize, simplify) $y^2 + x^3y = x^5z - 7x^3z^3 - 16x^2z^4 - 15xz^5 - 5z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 28x^3 - 64x^2 - 60x - 20$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, -15, -16, -7, 0, 1]), R([0, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, -15, -16, -7, 0, 1], R![0, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-20, -60, -64, -28, 0, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$6845$$ $$=$$ $$5 \cdot 37^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$6845$$ $$=$$ $$5 \cdot 37^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$24$$ $$=$$ $$2^{3} \cdot 3$$ $$I_4$$ $$=$$ $$852$$ $$=$$ $$2^{2} \cdot 3 \cdot 71$$ $$I_6$$ $$=$$ $$-14064$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 293$$ $$I_{10}$$ $$=$$ $$-27380$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 37^{2}$$ $$J_2$$ $$=$$ $$12$$ $$=$$ $$2^{2} \cdot 3$$ $$J_4$$ $$=$$ $$-136$$ $$=$$ $$- 2^{3} \cdot 17$$ $$J_6$$ $$=$$ $$2040$$ $$=$$ $$2^{3} \cdot 3 \cdot 5 \cdot 17$$ $$J_8$$ $$=$$ $$1496$$ $$=$$ $$2^{3} \cdot 11 \cdot 17$$ $$J_{10}$$ $$=$$ $$-6845$$ $$=$$ $$- 5 \cdot 37^{2}$$ $$g_1$$ $$=$$ $$-248832/6845$$ $$g_2$$ $$=$$ $$235008/6845$$ $$g_3$$ $$=$$ $$-58752/1369$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

Known points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(-1 : 0 : 1)$$ $$(-1 : 1 : 1)$$ $$(-2 : 3 : 1)$$ $$(-2 : 5 : 1)$$
$$(3 : -7 : 1)$$ $$(-3 : 7 : 4)$$ $$(3 : -20 : 1)$$ $$(-3 : 20 : 4)$$

magma: [C![-3,7,4],C![-3,20,4],C![-2,3,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![1,-1,0],C![1,0,0],C![3,-20,1],C![3,-7,1]];

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
$$(-1 : 0 : 1) - (1 : -1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.220557$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + 3xz + 3z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 - 4z^3$$ $$0.102222$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.022546$$ Real period: $$14.66307$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.330594$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 5 T^{2} )$$
$$37$$ $$2$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 37.a1
Elliptic curve 185.b1

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.