Properties

 Label 8450.c.84500.1 Conductor 8450 Discriminant 84500 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

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -5, 10, -5], R![1, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -5, 10, -5]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = -5x^4 + 10x^3 - 5x^2$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$8450$$ = $$2 \cdot 5^{2} \cdot 13^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$84500$$ = $$2^{2} \cdot 5^{3} \cdot 13^{2}$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$3944$$ = $$2^{3} \cdot 17 \cdot 29$$ $$I_4$$ = $$243556$$ = $$2^{2} \cdot 60889$$ $$I_6$$ = $$286158056$$ = $$2^{3} \cdot 35769757$$ $$I_{10}$$ = $$346112000$$ = $$2^{14} \cdot 5^{3} \cdot 13^{2}$$ $$J_2$$ = $$493$$ = $$17 \cdot 29$$ $$J_4$$ = $$7590$$ = $$2 \cdot 3 \cdot 5 \cdot 11 \cdot 23$$ $$J_6$$ = $$128000$$ = $$2^{10} \cdot 5^{3}$$ $$J_8$$ = $$1373975$$ = $$5^{2} \cdot 54959$$ $$J_{10}$$ = $$84500$$ = $$2^{2} \cdot 5^{3} \cdot 13^{2}$$ $$g_1$$ = $$29122898485693/84500$$ $$g_2$$ = $$90945776163/8450$$ $$g_3$$ = $$62220544/169$$
Alternative geometric invariants: G2

Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

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 everywhere.

magma: [C![0,-1,1],C![0,0,1],C![1,-18,3],C![1,-10,3],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,-4,1],C![3,-18,1],C![3,-10,1]];

Known rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : -18 : 3), (1 : -10 : 3), (1 : -5 : 2), (1 : -4 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (2 : -5 : 1), (2 : -4 : 1), (3 : -18 : 1), (3 : -10 : 1)

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

Number of rational Weierstrass points: $$0$$

Invariants of the Jacobian:

Analytic rank*: $$2$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.109881447913 Real period: 20.369950202679124490548652016 Tamagawa numbers: 2 (p = 2), 3 (p = 5), 1 (p = 13) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{6}\Z$$

Sato-Tate group

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

Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 65.a1
Elliptic curve 130.a2

Endomorphisms

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 $$\Q$$.