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: Magma / SageMath

Simplified equation

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

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

 $y^2 = x^6 + 4x^4 + 4x^2 + 1$ (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)

Invariants

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

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-7936$$ = $$- 2^{8} \cdot 31$$ $$I_4$$ = $$185344$$ = $$2^{10} \cdot 181$$ $$I_6$$ = $$-487358464$$ = $$- 2^{15} \cdot 107 \cdot 139$$ $$I_{10}$$ = $$-1677721600$$ = $$- 2^{26} \cdot 5^{2}$$ $$J_2$$ = $$-992$$ = $$- 2^{5} \cdot 31$$ $$J_4$$ = $$39072$$ = $$2^{5} \cdot 3 \cdot 11 \cdot 37$$ $$J_6$$ = $$-1945600$$ = $$- 2^{12} \cdot 5^{2} \cdot 19$$ $$J_8$$ = $$100853504$$ = $$2^{8} \cdot 151 \cdot 2609$$ $$J_{10}$$ = $$-409600$$ = $$- 2^{14} \cdot 5^{2}$$ $$g_1$$ = $$58632501248/25$$ $$g_2$$ = $$2327987904/25$$ $$g_3$$ = $$4674304$$

Automorphism group

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

Rational points

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

Points: $$(0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -1 : 0),\, (1 : 1 : 0)$$

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

Number of rational Weierstrass points: $$0$$

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.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));

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

Generator Height Order
$$x^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0$$ $$3$$
$$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$6$$

BSD invariants

 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
$$2$$ $$14$$ $$4$$ $$9$$ $$1$$
$$5$$ $$2$$ $$2$$ $$1$$ $$( 1 + T )^{2}$$

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 $$\Q$$.