# Properties

 Label 6400.f.64000.1 Conductor 6400 Discriminant -64000 Sato-Tate group $E_4$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 6, 1, -3, -2], R![0, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 6, 1, -3, -2]), R([0, 0, 0, 1]))

$y^2 + x^3y = -2x^4 - 3x^3 + x^2 + 6x + 4$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$6400$$ = $$2^{8} \cdot 5^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-64000$$ = $$-1 \cdot 2^{9} \cdot 5^{3}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-2464$$ = $$-1 \cdot 2^{5} \cdot 7 \cdot 11$$ $$I_4$$ = $$79360$$ = $$2^{9} \cdot 5 \cdot 31$$ $$I_6$$ = $$-79790080$$ = $$-1 \cdot 2^{15} \cdot 5 \cdot 487$$ $$I_{10}$$ = $$-262144000$$ = $$-1 \cdot 2^{21} \cdot 5^{3}$$ $$J_2$$ = $$-308$$ = $$-1 \cdot 2^{2} \cdot 7 \cdot 11$$ $$J_4$$ = $$3126$$ = $$2 \cdot 3 \cdot 521$$ $$J_6$$ = $$164$$ = $$2^{2} \cdot 41$$ $$J_8$$ = $$-2455597$$ = $$-1 \cdot 367 \cdot 6691$$ $$J_{10}$$ = $$-64000$$ = $$-1 \cdot 2^{9} \cdot 5^{3}$$ $$g_1$$ = $$5413568314/125$$ $$g_2$$ = $$713561079/500$$ $$g_3$$ = $$-243089/1000$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_4$$ (GAP id : [4,1]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$D_4$$ (GAP id : [8,3])

## 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![-17,2385,12],C![-17,2528,12],C![-3,11,2],C![-3,16,2],C![-2,2,1],C![-2,6,1],C![-1,0,1],C![-1,1,1],C![0,-2,1],C![0,2,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![7,-243,5],C![7,-100,5]];

Known rational points: (-17 : 2385 : 12), (-17 : 2528 : 12), (-3 : 11 : 2), (-3 : 16 : 2), (-2 : 2 : 1), (-2 : 6 : 1), (-1 : 0 : 1), (-1 : 1 : 1), (0 : -2 : 1), (0 : 2 : 1), (1 : -3 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 2 : 1), (7 : -243 : 5), (7 : -100 : 5)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$2$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$4$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0670321386023 Real period: 19.455210215850228941286556289 Tamagawa numbers: 2 (p = 2), 2 (p = 5) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z \times \Z/{2}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_4$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

### Decomposition

Splits over the number field $$\Q (b) \simeq$$ 4.4.256000.1 with defining polynomial:
$$x^{4} - 20 x^{2} + 10$$

Decomposes up to isogeny as the square of the elliptic curve:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = \frac{1314223139840}{9464869683} b^{3} + \frac{5861441561600}{9464869683} b^{2} - \frac{276644126720}{9464869683} b - \frac{2349793825280}{9464869683}$$
$$g_6 = -\frac{7920508872718008320}{531632265224427} b^{3} - \frac{35011457079460659200}{531632265224427} b^{2} + \frac{3787394125533593600}{531632265224427} b + \frac{17636598900106035200}{531632265224427}$$
Conductor norm: 1

### Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ 4.4.256000.1 with defining polynomial $$x^{4} - 20 x^{2} + 10$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

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

### Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{10})$$ with generator $$\frac{1}{3} a^{2} - \frac{10}{3}$$ with minimal polynomial $$x^{2} - 10$$:
 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\sqrt{-1}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-1})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: $E_2$
of $$\GL_2$$-type, simple