Properties

Label 20172.b.968256.1
Conductor 20172
Discriminant -968256
Mordell-Weil group \(\Z \times \Z \times \Z/{2}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^6 - x^5 - 2x^4 + 4x^3 - 3x + 1$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^6 - x^5z - 2x^4z^2 + 4x^3z^3 - 3xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 - 4x^5 - 7x^4 + 18x^3 + x^2 - 12x + 4$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -3, 0, 4, -2, -1, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -3, 0, 4, -2, -1, 1]), R([0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([4, -12, 1, 18, -7, -4, 4]))
 

Invariants

Conductor: \( N \)  =  \(20172\) = \( 2^{2} \cdot 3 \cdot 41^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-968256\) = \( - 2^{6} \cdot 3^{2} \cdot 41^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(136\) =  \( 2^{3} \cdot 17 \)
\( I_4 \)  = \(774916\) =  \( 2^{2} \cdot 23 \cdot 8423 \)
\( I_6 \)  = \(-219321592\) =  \( - 2^{3} \cdot 7 \cdot 613 \cdot 6389 \)
\( I_{10} \)  = \(-3965976576\) =  \( - 2^{18} \cdot 3^{2} \cdot 41^{2} \)
\( J_2 \)  = \(17\) =  \( 17 \)
\( J_4 \)  = \(-8060\) =  \( - 2^{2} \cdot 5 \cdot 13 \cdot 31 \)
\( J_6 \)  = \(418896\) =  \( 2^{4} \cdot 3^{2} \cdot 2909 \)
\( J_8 \)  = \(-14460592\) =  \( - 2^{4} \cdot 83 \cdot 10889 \)
\( J_{10} \)  = \(-968256\) =  \( - 2^{6} \cdot 3^{2} \cdot 41^{2} \)
\( g_1 \)  = \(-1419857/968256\)
\( g_2 \)  = \(9899695/242064\)
\( g_3 \)  = \(-840701/6724\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\)
\((1 : -2 : 1)\) \((1 : -3 : 2)\) \((2 : 3 : 1)\) \((-3 : 5 : 2)\) \((2 : -9 : 1)\) \((-3 : -11 : 2)\)
\((3 : -52 : 5)\) \((3 : -68 : 5)\)

magma: [C![-3,-11,2],C![-3,5,2],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,1],C![1,1,0],C![2,-9,1],C![2,3,1],C![3,-68,5],C![3,-52,5]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z \times \Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0.158713\) \(\infty\)
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.112353\) \(\infty\)
\((-1 : 0 : 1) + (1 : -3 : 2) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x + z) (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)

2-torsion field: 4.0.656.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.017831 \)
Real period: \( 16.48290 \)
Tamagawa product: \( 8 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.587845 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(6\) \(2\) \(4\) \(( 1 + T )^{2}\)
\(3\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(41\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

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 246.a1
  Elliptic curve 82.a2

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 \(\Q\).