Properties

 Label 1197.a.410571.1 Conductor 1197 Discriminant -410571 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

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -3, -7, 12, 1], R![1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -3, -7, 12, 1]), R([1, 0, 1]))

$y^2 + (x^2 + 1)y = x^5 + 12x^4 - 7x^3 - 3x^2 + x$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1197$$ = $$3^{2} \cdot 7 \cdot 19$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-410571$$ = $$-1 \cdot 3^{2} \cdot 7^{4} \cdot 19$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$13184$$ = $$2^{7} \cdot 103$$ $$I_4$$ = $$11308480$$ = $$2^{6} \cdot 5 \cdot 35339$$ $$I_6$$ = $$37014592960$$ = $$2^{6} \cdot 5 \cdot 8699 \cdot 13297$$ $$I_{10}$$ = $$-1681698816$$ = $$-1 \cdot 2^{12} \cdot 3^{2} \cdot 7^{4} \cdot 19$$ $$J_2$$ = $$1648$$ = $$2^{4} \cdot 103$$ $$J_4$$ = $$-4634$$ = $$-1 \cdot 2 \cdot 7 \cdot 331$$ $$J_6$$ = $$23921$$ = $$19 \cdot 1259$$ $$J_8$$ = $$4486963$$ = $$13 \cdot 17 \cdot 79 \cdot 257$$ $$J_{10}$$ = $$-410571$$ = $$-1 \cdot 3^{2} \cdot 7^{4} \cdot 19$$ $$g_1$$ = $$-12155869717331968/410571$$ $$g_2$$ = $$2962986082304/58653$$ $$g_3$$ = $$-3419323136/21609$$
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 everywhere.

magma: [C![-1,-34,4],C![0,-1,1],C![0,0,1],C![1,0,0]];

All rational points: (-1 : -34 : 4), (0 : -1 : 1), (0 : 0 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$2$$

Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 3), 2 (p = 7), 1 (p = 19) 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 $$\Q$$.