Properties

Label 1122.a.1122.1
Conductor 1122
Discriminant 1122
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
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

$y^2 + (x^2 + x)y = x^5 + 7x^4 - 43x^2 + 51x - 17$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + 7x^4z^2 - 43x^2z^4 + 51xz^5 - 17z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 29x^4 + 2x^3 - 171x^2 + 204x - 68$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-17, 51, -43, 0, 7, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-17, 51, -43, 0, 7, 1]), R([0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-68, 204, -171, 2, 29, 4]))
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(112008\) =  \( 2^{3} \cdot 3 \cdot 13 \cdot 359 \)
\( I_4 \)  = \(1153284\) =  \( 2^{2} \cdot 3 \cdot 11 \cdot 8737 \)
\( I_6 \)  = \(42651340296\) =  \( 2^{3} \cdot 3 \cdot 19 \cdot 93533641 \)
\( I_{10} \)  = \(4595712\) =  \( 2^{13} \cdot 3 \cdot 11 \cdot 17 \)
\( J_2 \)  = \(14001\) =  \( 3 \cdot 13 \cdot 359 \)
\( J_4 \)  = \(8155820\) =  \( 2^{2} \cdot 5 \cdot 407791 \)
\( J_6 \)  = \(6325887612\) =  \( 2^{2} \cdot 3 \cdot 11 \cdot 17 \cdot 2819023 \)
\( J_8 \)  = \(5512838145803\) =  \( 198173 \cdot 27818311 \)
\( J_{10} \)  = \(1122\) =  \( 2 \cdot 3 \cdot 11 \cdot 17 \)
\( g_1 \)  = \(179338702480653356667/374\)
\( g_2 \)  = \(3730727674118765970/187\)
\( g_3 \)  = \(1105214886926046\)

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

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

magma: [C![1,-1,1],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: \(\Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(4x^2 + 17xz - 17z^2\) \(=\) \(0,\) \(8y\) \(=\) \(13xz^2 - 17z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + 4xz - 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(3xz^2 - 4z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{561})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 6.820718 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.426294 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 66.b2
  Elliptic curve 17.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 \(\Q\).