Properties

Label 799680.b.799680.1
Conductor 799680
Discriminant -799680
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![446, 0, 336, 0, 84, 0, 7], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([446, 0, 336, 0, 84, 0, 7]), R([1, 0, 1]))
 

$y^2 + (x^2 + 1)y = 7x^6 + 84x^4 + 336x^2 + 446$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 799680 \)  =  \( 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-799680\)  =  \( -1 \cdot 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \)

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);
 

\( I_2 \)  =  \(-19252832\)  =  \( -1 \cdot 2^{5} \cdot 601651 \)
\( I_4 \)  =  \(228856000\)  =  \( 2^{6} \cdot 5^{3} \cdot 28607 \)
\( I_6 \)  =  \(-1467556505638912\)  =  \( -1 \cdot 2^{11} \cdot 13 \cdot 163 \cdot 338169101 \)
\( I_{10} \)  =  \(-3275489280\)  =  \( -1 \cdot 2^{18} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \)
\( J_2 \)  =  \(-2406604\)  =  \( -1 \cdot 2^{2} \cdot 601651 \)
\( J_4 \)  =  \(241320233284\)  =  \( 2^{2} \cdot 13 \cdot 4640773717 \)
\( J_6 \)  =  \(-32263933356762240\)  =  \( -1 \cdot 2^{7} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \cdot 89 \cdot 151 \cdot 1501081 \)
\( J_8 \)  =  \(4852764019968313102076\)  =  \( 2^{2} \cdot 41 \cdot 47 \cdot 127 \cdot 632911 \cdot 7832512601 \)
\( J_{10} \)  =  \(-799680\)  =  \( -1 \cdot 2^{6} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 17 \)
\( g_1 \)  =  \(1261372031256529020641523732016/12495\)
\( g_2 \)  =  \(52556648780600635811581759084/12495\)
\( g_3 \)  =  \(233673974755154692792288\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

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: [];
 

No rational points known.

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(1\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
 

2-Selmer rank: \(6\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 2.30612468166

Real period: 2.0358049513911927833065139047

Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 7), 1 (p = 17)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

Torsion: \(\Z/{6}\Z\)

2-torsion field: splitting field of \(x^{8} - 4 x^{7} + 90 x^{6} - 200 x^{5} + 2123 x^{4} - 2480 x^{3} + 14610 x^{2} + 3668 x + 10306\) with Galois group $D_4\times C_2$

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 57120.bd1
  Elliptic curve 14.a4

Endomorphisms

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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