Properties

Label 19881.b.536787.1
Conductor 19881
Discriminant -536787
Mordell-Weil group \(\Z \times \Z\)
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

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^6 - 3x^5 + 2x^4 + x^3 - 2x^2 + x$ (homogenize, simplify)
$y^2 + z^3y = x^6 - 3x^5z + 2x^4z^2 + x^3z^3 - 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 12x^5 + 8x^4 + 4x^3 - 8x^2 + 4x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -2, 1, 2, -3, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -2, 1, 2, -3, 1]), R([1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 4, -8, 4, 8, -12, 4]))
 

Invariants

Conductor: \( N \)  =  \(19881\) = \( 3^{2} \cdot 47^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-536787\) = \( - 3^{5} \cdot 47^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-1760\) =  \( - 2^{5} \cdot 5 \cdot 11 \)
\( I_4 \)  = \(16192\) =  \( 2^{6} \cdot 11 \cdot 23 \)
\( I_6 \)  = \(-27226112\) =  \( - 2^{12} \cdot 17^{2} \cdot 23 \)
\( I_{10} \)  = \(-2198679552\) =  \( - 2^{12} \cdot 3^{5} \cdot 47^{2} \)
\( J_2 \)  = \(-220\) =  \( - 2^{2} \cdot 5 \cdot 11 \)
\( J_4 \)  = \(1848\) =  \( 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
\( J_6 \)  = \(12312\) =  \( 2^{3} \cdot 3^{4} \cdot 19 \)
\( J_8 \)  = \(-1530936\) =  \( - 2^{3} \cdot 3^{2} \cdot 11 \cdot 1933 \)
\( J_{10} \)  = \(-536787\) =  \( - 3^{5} \cdot 47^{2} \)
\( g_1 \)  = \(515363200000/536787\)
\( g_2 \)  = \(6559168000/178929\)
\( g_3 \)  = \(-7356800/6627\)

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

Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : 1 : 1)\)
\((1 : -1 : 1)\) \((-1 : -2 : 1)\) \((2 : 1 : 1)\) \((2 : -2 : 1)\) \((-1 : -164 : 11)\) \((12 : -164 : 11)\)
\((-1 : -1167 : 11)\) \((12 : -1167 : 11)\)

magma: [C![-1,-1167,11],C![-1,-164,11],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-2,1],C![2,1,1],C![12,-1167,11],C![12,-164,11]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((-x + 2z) x\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.116491\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.068973\) \(\infty\)

2-torsion field: 6.4.3817152.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.006845 \)
Real period: \( 14.07745 \)
Tamagawa product: \( 7 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.674566 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(5\) \(2\) \(7\) \(( 1 - T )( 1 + T )\)
\(47\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

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 141.a1
  Elliptic curve 141.d1

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 \(\Q\).