# Properties

 Label 400.a.409600.1 Conductor 400 Discriminant -409600 Mordell-Weil group $$\Z/{3}\Z \times \Z/{6}\Z$$ Sato-Tate group $E_1$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: SageMath / Magma

## Simplified equation

 $y^2 = x^6 + 4x^4 + 4x^2 + 1$ (homogenize, simplify) $y^2 = x^6 + 4x^4z^2 + 4x^2z^4 + z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^4 + 4x^2 + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 0, 4, 0, 4, 0, 1]), R([]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 0, 4, 0, 4, 0, 1], R![]);

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

magma: X,pi:= SimplifiedModel(C);

## Invariants

 Conductor: $$N$$ $$=$$ $$400$$ $$=$$ $$2^{4} \cdot 5^{2}$$ magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(400,2),R![1]>*])); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-409600$$ $$=$$ $$- 2^{14} \cdot 5^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ $$=$$ $$248$$ $$=$$ $$2^{3} \cdot 31$$ $$I_4$$ $$=$$ $$181$$ $$=$$ $$181$$ $$I_6$$ $$=$$ $$14873$$ $$=$$ $$107 \cdot 139$$ $$I_{10}$$ $$=$$ $$50$$ $$=$$ $$2 \cdot 5^{2}$$ $$J_2$$ $$=$$ $$992$$ $$=$$ $$2^{5} \cdot 31$$ $$J_4$$ $$=$$ $$39072$$ $$=$$ $$2^{5} \cdot 3 \cdot 11 \cdot 37$$ $$J_6$$ $$=$$ $$1945600$$ $$=$$ $$2^{12} \cdot 5^{2} \cdot 19$$ $$J_8$$ $$=$$ $$100853504$$ $$=$$ $$2^{8} \cdot 151 \cdot 2609$$ $$J_{10}$$ $$=$$ $$409600$$ $$=$$ $$2^{14} \cdot 5^{2}$$ $$g_1$$ $$=$$ $$58632501248/25$$ $$g_2$$ $$=$$ $$2327987904/25$$ $$g_3$$ $$=$$ $$4674304$$

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

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

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $D_4$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$D_4$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

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

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

Number of rational Weierstrass points: $$0$$

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: $$\Z/{3}\Z \times \Z/{6}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$3$$
$$(0 : -1 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$6$$

## BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$7.977094$$ Tamagawa product: $$9$$ Torsion order: $$18$$ Leading coefficient: $$0.221585$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$14$$ $$9$$ $$1$$
$$5$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_1$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 20.a3

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ a non-Eichler order of index $$4$$ in a maximal order of $$\End (J_{}) \otimes \Q$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

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