# Properties

 Label 971.a.971.1 Conductor 971 Discriminant -971 Mordell-Weil group $$\Z$$ 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

 $y^2 + y = x^5 - 2x^3 + x$ (homogenize, simplify) $y^2 + z^3y = x^5z - 2x^3z^3 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 8x^3 + 4x + 1$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, 4, 0, -8, 0, 4]))

## Invariants

 Conductor: $$N$$ = $$971$$ = $$971$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-971$$ = $$- 971$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$1024$$ = $$2^{10}$$ $$I_4$$ = $$16384$$ = $$2^{14}$$ $$I_6$$ = $$5139456$$ = $$2^{10} \cdot 3 \cdot 7 \cdot 239$$ $$I_{10}$$ = $$-3977216$$ = $$- 2^{12} \cdot 971$$ $$J_2$$ = $$128$$ = $$2^{7}$$ $$J_4$$ = $$512$$ = $$2^{9}$$ $$J_6$$ = $$2000$$ = $$2^{4} \cdot 5^{3}$$ $$J_8$$ = $$-1536$$ = $$- 2^{9} \cdot 3$$ $$J_{10}$$ = $$-971$$ = $$- 971$$ $$g_1$$ = $$-34359738368/971$$ $$g_2$$ = $$-1073741824/971$$ $$g_3$$ = $$-32768000/971$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## Automorphism group

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

## Rational points

All points: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (-1 : -1 : 1),\, (1 : -1 : 1)$$

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

Number of rational Weierstrass points: $$1$$

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

This curve is locally solvable everywhere.

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);

## Mordell-Weil group of the Jacobian:

Group structure: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(x - z) (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0.005970$$ $$\infty$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.005970$$ Real period: $$29.64711$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.176998$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$971$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 45 T + 971 T^{2} )$$

## 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 $$\Q$$.