Properties

Label 799680.b.799680.1
Conductor 799680
Discriminant -799680
Mordell-Weil group \(\Z \times \Z/{6}\Z\)
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

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = 7x^6 + 84x^4 + 336x^2 + 446$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = 7x^6 + 84x^4z^2 + 336x^2z^4 + 446z^6$ (dehomogenize, simplify)
$y^2 = 28x^6 + 337x^4 + 1346x^2 + 1785$ (minimize, homogenize)

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]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1785, 0, 1346, 0, 337, 0, 28]))
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

No rational points are known for this curve.

magma: [];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + 2z^3\) \(2.306124\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 + 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(2z^3\) \(0\) \(6\)

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 $D_4\times C_2$

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(6\)
Regulator: \( 2.306124 \)
Real period: \( 2.035804 \)
Tamagawa product: \( 1 \)
Torsion order:\( 6 \)
Leading coefficient: \( 2.086586 \)
Analytic order of Ш: \( 16 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(6\) \(6\) \(1\) \(1 + T\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 5 T^{2} )\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )^{2}\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 17 T^{2} )\)

Sato-Tate group

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

Decomposition of the Jacobian

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 the Jacobian

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 \(\Q\).