Properties

 Label 1269.b.102789.1 Conductor 1269 Discriminant -102789 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

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-103, -16, -80, -8, -21, -1, -2], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-103, -16, -80, -8, -21, -1, -2]), R([0, 1, 0, 1]))

$y^2 + (x^3 + x)y = -2x^6 - x^5 - 21x^4 - 8x^3 - 80x^2 - 16x - 103$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1269$$ = $$3^{3} \cdot 47$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-102789$$ = $$-1 \cdot 3^{7} \cdot 47$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-1094304$$ = $$-1 \cdot 2^{5} \cdot 3 \cdot 11399$$ $$I_4$$ = $$2865600$$ = $$2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 199$$ $$I_6$$ = $$-1043681025600$$ = $$-1 \cdot 2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 24159283$$ $$I_{10}$$ = $$-421023744$$ = $$-1 \cdot 2^{12} \cdot 3^{7} \cdot 47$$ $$J_2$$ = $$-136788$$ = $$-1 \cdot 2^{2} \cdot 3 \cdot 11399$$ $$J_4$$ = $$779593356$$ = $$2^{2} \cdot 3^{3} \cdot 7218457$$ $$J_6$$ = $$-5923938871071$$ = $$-1 \cdot 3^{4} \cdot 13 \cdot 52567 \cdot 107021$$ $$J_8$$ = $$50639487394179303$$ = $$3^{5} \cdot 163 \cdot 149561 \cdot 8548247$$ $$J_{10}$$ = $$-102789$$ = $$-1 \cdot 3^{7} \cdot 47$$ $$g_1$$ = $$197075993647247827966976/423$$ $$g_2$$ = $$2737061778548953841408/141$$ $$g_3$$ = $$152047414479420367856/141$$
Alternative geometric invariants: G2

Automorphism group

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

Rational points

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 $\R$.

magma: [];

There are no rational points.

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

Number of rational Weierstrass points: $$0$$

Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: twice a square Tamagawa numbers: 5 (p = 3), 1 (p = 47) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

Decomposition

Simple over $$\overline{\Q}$$

Endomorphisms

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 $$\Q$$.