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

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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);
 
\( \Delta \)  =  \(-64000\)  =  \( -1 \cdot 2^{9} \cdot 5^{3} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

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: Igusa-Clebsch, Igusa, 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: \(\Z/{2}\Z \times \Z/{2}\Z\)

2-torsion field: 8.0.40960000.1

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