Properties

Label 1088.b.2176.1
Conductor 1088
Discriminant -2176
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $N(G_{1,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM} \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = 4x^4 + 24x^2 + 34$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 4x^4z^2 + 24x^2z^4 + 34z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 18x^4 + 97x^2 + 136$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([34, 0, 24, 0, 4]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![34, 0, 24, 0, 4], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([136, 0, 97, 0, 18, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(1088\) = \( 2^{6} \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-2176\) = \( - 2^{7} \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-60576\) =  \( - 2^{5} \cdot 3 \cdot 631 \)
\( I_4 \)  = \(4359360\) =  \( 2^{6} \cdot 3 \cdot 5 \cdot 19 \cdot 239 \)
\( I_6 \)  = \(-84995229696\) =  \( - 2^{11} \cdot 3 \cdot 13 \cdot 83 \cdot 12821 \)
\( I_{10} \)  = \(-8912896\) =  \( - 2^{19} \cdot 17 \)
\( J_2 \)  = \(-7572\) =  \( - 2^{2} \cdot 3 \cdot 631 \)
\( J_4 \)  = \(2343556\) =  \( 2^{2} \cdot 585889 \)
\( J_6 \)  = \(-952909568\) =  \( - 2^{8} \cdot 13 \cdot 17 \cdot 16843 \)
\( J_8 \)  = \(430794130940\) =  \( 2^{2} \cdot 5 \cdot 37 \cdot 9467 \cdot 61493 \)
\( J_{10} \)  = \(-2176\) =  \( - 2^{7} \cdot 17 \)
\( g_1 \)  = \(194465720403941544/17\)
\( g_2 \)  = \(7948719687495546/17\)
\( g_3 \)  = \(25108109106912\)

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

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

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
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(6\)

2-torsion field: 8.0.18939904.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 5.893943 \)
Tamagawa product: \( 3 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.491161 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(7\) \(6\) \(3\) \(1 - T\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 17 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(G_{1,3})$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 32.a1
  Elliptic curve 34.a3

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(8\) in \(\Z \times \Z [\sqrt{-1}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-1}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)