Properties

 Label 961.a.923521.1 Conductor 961 Discriminant 923521 Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

Related objects

Show commands for: Magma / SageMath

This is a model for the modular curve $X_0(31)$.

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -2, 3, 4, -5], R![1, 0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -2, 3, 4, -5]), R([1, 0, 1, 1]))

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

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$961$$ = $$31^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$923521$$ = $$31^{4}$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$8200$$ = $$2^{3} \cdot 5^{2} \cdot 41$$ $$I_4$$ = $$315844$$ = $$2^{2} \cdot 281^{2}$$ $$I_6$$ = $$753213512$$ = $$2^{3} \cdot 94151689$$ $$I_{10}$$ = $$3782742016$$ = $$2^{12} \cdot 31^{4}$$ $$J_2$$ = $$1025$$ = $$5^{2} \cdot 41$$ $$J_4$$ = $$40486$$ = $$2 \cdot 31 \cdot 653$$ $$J_6$$ = $$2121888$$ = $$2^{5} \cdot 3 \cdot 23 \cdot 31^{2}$$ $$J_8$$ = $$133954751$$ = $$7 \cdot 31^{2} \cdot 19913$$ $$J_{10}$$ = $$923521$$ = $$31^{4}$$ $$g_1$$ = $$1131408212890625/923521$$ $$g_2$$ = $$1406419156250/29791$$ $$g_3$$ = $$2319780000/961$$
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,-1,0],C![1,0,0]];

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

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: $$0$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 5 (p = 31) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{5}\Z$$

Sato-Tate group

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

Decomposition

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

Endomorphisms

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{5}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{5})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.