# Properties

 Label 100293.a.902637.1 Conductor 100293 Discriminant -902637 Mordell-Weil group $$\Z/{3}\Z$$ 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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 2, -7, 3, -5, 1, -1], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 2, -7, 3, -5, 1, -1]), R([1, 1, 0, 1]))

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 2, -7, 3, -5, 1, -1], R![1, 1, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-7, 10, -27, 14, -18, 4, -3]))

 $y^2 + (x^3 + x + 1)y = -x^6 + x^5 - 5x^4 + 3x^3 - 7x^2 + 2x - 2$ (homogenize, simplify) $y^2 + (x^3 + xz^2 + z^3)y = -x^6 + x^5z - 5x^4z^2 + 3x^3z^3 - 7x^2z^4 + 2xz^5 - 2z^6$ (dehomogenize, simplify) $y^2 = -3x^6 + 4x^5 - 18x^4 + 14x^3 - 27x^2 + 10x - 7$ (minimize, homogenize)

## Invariants

 $$N$$ = $$100293$$ = $$3 \cdot 101 \cdot 331$$ magma: Conductor(LSeries(C)); Factorization($1); $$\Delta$$ = $$-902637$$ = $$- 3^{3} \cdot 101 \cdot 331$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-10040$$ = $$- 2^{3} \cdot 5 \cdot 251$$ $$I_4$$ = $$602980$$ = $$2^{2} \cdot 5 \cdot 7 \cdot 59 \cdot 73$$ $$I_6$$ = $$-1884167064$$ = $$- 2^{3} \cdot 3^{2} \cdot 13 \cdot 79 \cdot 83 \cdot 307$$ $$I_{10}$$ = $$-3697201152$$ = $$- 2^{12} \cdot 3^{3} \cdot 101 \cdot 331$$ $$J_2$$ = $$-1255$$ = $$- 5 \cdot 251$$ $$J_4$$ = $$59345$$ = $$5 \cdot 11 \cdot 13 \cdot 83$$ $$J_6$$ = $$-3494111$$ = $$- 1223 \cdot 2857$$ $$J_8$$ = $$215820070$$ = $$2 \cdot 5 \cdot 107 \cdot 201701$$ $$J_{10}$$ = $$-902637$$ = $$- 3^{3} \cdot 101 \cdot 331$$ $$g_1$$ = $$3113283207034375/902637$$ $$g_2$$ = $$117304672574375/902637$$ $$g_3$$ = $$5503312177775/902637$$

## Automorphism group

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

## 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 $\R$.

## Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

Group structure: $$\Z/{3}\Z$$

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

## BSD invariants

 Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$4.422795$$ Tamagawa product: $$3$$ Torsion order: $$3$$ Leading coefficient: $$2.948530$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$3$$ $$3$$ $$1$$ $$3$$ $$( 1 - T )( 1 + T + 3 T^{2} )$$
$$101$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 101 T^{2} )$$
$$331$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 5 T + 331 T^{2} )$$

## 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 $$\Q$$.