Properties

Label 1122.b.2244.1
Conductor 1122
Discriminant 2244
Mordell-Weil group \(\Z/{2}\Z \times \Z/{6}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^5 + 7x^4 + 5x^3 - x^2 - x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + 7x^4z^2 + 5x^3z^3 - x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 29x^4 + 22x^3 - 3x^2 - 4x$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 5, 7, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 5, 7, 1]), R([0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([0, -4, -3, 22, 29, 4]))
 

Invariants

Conductor: \( N \)  =  \(1122\) = \( 2 \cdot 3 \cdot 11 \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(2244\) = \( 2^{2} \cdot 3 \cdot 11 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(3656\) =  \( 2^{3} \cdot 457 \)
\( I_4 \)  = \(615172\) =  \( 2^{2} \cdot 113 \cdot 1361 \)
\( I_6 \)  = \(590801160\) =  \( 2^{3} \cdot 3 \cdot 5 \cdot 349 \cdot 14107 \)
\( I_{10} \)  = \(9191424\) =  \( 2^{14} \cdot 3 \cdot 11 \cdot 17 \)
\( J_2 \)  = \(457\) =  \( 457 \)
\( J_4 \)  = \(2294\) =  \( 2 \cdot 31 \cdot 37 \)
\( J_6 \)  = \(8704\) =  \( 2^{9} \cdot 17 \)
\( J_8 \)  = \(-321177\) =  \( - 3 \cdot 151 \cdot 709 \)
\( J_{10} \)  = \(2244\) =  \( 2^{2} \cdot 3 \cdot 11 \cdot 17 \)
\( g_1 \)  = \(19933382494057/2244\)
\( g_2 \)  = \(109474259971/1122\)
\( g_3 \)  = \(26732672/33\)

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : -1 : 1),\, (-1 : 1 : 1)\)

magma: [C![-1,-1,1],C![-1,1,1],C![0,0,1],C![1,0,0]];
 

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: \(\Z/{2}\Z \times \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 + xz - z^2\) \(=\) \(0,\) \(8y\) \(=\) \(-3xz^2 - z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(3xz^2 + 2z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{17}, \sqrt{33})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 30.72413 \)
Tamagawa product: \( 2 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.426724 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 - T + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 11 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 17 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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