# Properties

 Label 847.d.456533.1 Conductor 847 Discriminant 456533 Mordell-Weil group $$\Z/{15}\Z$$ 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: Magma / SageMath

## Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, -24, 37, 3, -22, -9, -1], R![1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, -24, 37, 3, -22, -9, -1]), R([1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, -24, 37, 3, -22, -9, -1], R![1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([17, -96, 148, 12, -88, -36, -4]))

 $y^2 + y = -x^6 - 9x^5 - 22x^4 + 3x^3 + 37x^2 - 24x + 4$ (homogenize, simplify) $y^2 + z^3y = -x^6 - 9x^5z - 22x^4z^2 + 3x^3z^3 + 37x^2z^4 - 24xz^5 + 4z^6$ (dehomogenize, simplify) $y^2 = -4x^6 - 36x^5 - 88x^4 + 12x^3 + 148x^2 - 96x + 17$ (minimize, homogenize)

## Invariants

 $$N$$ = $$847$$ = $$7 \cdot 11^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$456533$$ = $$7^{3} \cdot 11^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$363808$$ = $$2^{5} \cdot 11369$$ $$I_4$$ = $$162112$$ = $$2^{6} \cdot 17 \cdot 149$$ $$I_6$$ = $$19446212608$$ = $$2^{11} \cdot 9495221$$ $$I_{10}$$ = $$1869959168$$ = $$2^{12} \cdot 7^{3} \cdot 11^{3}$$ $$J_2$$ = $$45476$$ = $$2^{2} \cdot 11369$$ $$J_4$$ = $$86167752$$ = $$2^{3} \cdot 3 \cdot 11 \cdot 23^{2} \cdot 617$$ $$J_6$$ = $$217689875480$$ = $$2^{3} \cdot 5 \cdot 7^{3} \cdot 11^{2} \cdot 131129$$ $$J_8$$ = $$618695823148744$$ = $$2^{3} \cdot 11^{2} \cdot 571 \cdot 1119349523$$ $$J_{10}$$ = $$456533$$ = $$7^{3} \cdot 11^{3}$$ $$g_1$$ = $$194496275421254111077376/456533$$ $$g_2$$ = $$736713878289412204032/41503$$ $$g_3$$ = $$10847340081772160/11$$

## Automorphism group

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

## Rational points

magma: [];

This curve has no rational points.

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

Number of rational Weierstrass points: $$0$$

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 except over $\Q_{11}$.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{15}\Z$$

Generator Height Order
$$2x^2 + 5xz - 4z^2$$ $$=$$ $$0,$$ $$4y$$ $$=$$ $$xz^2 - 4z^3$$ $$0$$ $$15$$

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$9.829455$$ Tamagawa product: $$3$$ Torsion order: $$15$$ Leading coefficient: $$0.262118$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$7$$ $$3$$ $$1$$ $$3$$ $$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$11$$ $$3$$ $$2$$ $$1$$ $$( 1 - T )( 1 + T )$$

## 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 77.b2
Elliptic curve 11.a3

## 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 $$\Q$$.