Properties

Label 1088.a.1088.1
Conductor 1088
Discriminant -1088
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $N(G_{3,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + 2x^2 + x + 1$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^4z^2 + x^3z^3 + 2x^2z^4 + xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 7x^4 + 8x^3 + 11x^2 + 6x + 5$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 1, 2, 1, 1], R![1, 1, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 1, 2, 1, 1]), R([1, 1, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([5, 6, 11, 8, 7, 2, 1]))
 

Invariants

Conductor: \( N \)  =  \(1088\) = \( 2^{6} \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-1088\) = \( - 2^{6} \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-1568\) =  \( - 2^{5} \cdot 7^{2} \)
\( I_4 \)  = \(1792\) =  \( 2^{8} \cdot 7 \)
\( I_6 \)  = \(-323584\) =  \( - 2^{12} \cdot 79 \)
\( I_{10} \)  = \(-4456448\) =  \( - 2^{18} \cdot 17 \)
\( J_2 \)  = \(-196\) =  \( - 2^{2} \cdot 7^{2} \)
\( J_4 \)  = \(1582\) =  \( 2 \cdot 7 \cdot 113 \)
\( J_6 \)  = \(-17884\) =  \( - 2^{2} \cdot 17 \cdot 263 \)
\( J_8 \)  = \(250635\) =  \( 3 \cdot 5 \cdot 7^{2} \cdot 11 \cdot 31 \)
\( J_{10} \)  = \(-1088\) =  \( - 2^{6} \cdot 17 \)
\( g_1 \)  = \(4519603984/17\)
\( g_2 \)  = \(186120718/17\)
\( g_3 \)  = \(631463\)

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

Rational points

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

magma: [C![1,-1,0],C![1,0,0]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 4.0.272.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 15.72012 \)
Tamagawa product: \( 1 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.436670 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
  \(x^{2} - 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 2.2.8.1-17.2-a2
  Elliptic curve 2.2.8.1-17.1-a1

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)

Of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)