Properties

Label 1088.b.2176.2
Conductor $1088$
Discriminant $2176$
Mordell-Weil group \(\Z/{12}\Z\)
Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{CM} \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

Learn more

Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -5x^4 + 24x^2 - 34$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -5x^4z^2 + 24x^2z^4 - 34z^6$ (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, -5]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-34, 0, 24, 0, -5], 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

\( I_2 \)  \(=\) \(7572\) \(=\)  \( 2^{2} \cdot 3 \cdot 631 \)
\( I_4 \)  \(=\) \(68115\) \(=\)  \( 3 \cdot 5 \cdot 19 \cdot 239 \)
\( I_6 \)  \(=\) \(166006308\) \(=\)  \( 2^{2} \cdot 3 \cdot 13 \cdot 83 \cdot 12821 \)
\( I_{10} \)  \(=\) \(272\) \(=\)  \( 2^{4} \cdot 17 \)
\( J_2 \)  \(=\) \(7572\) \(=\)  \( 2^{2} \cdot 3 \cdot 631 \)
\( J_4 \)  \(=\) \(2343556\) \(=\)  \( 2^{2} \cdot 585889 \)
\( J_6 \)  \(=\) \(952909568\) \(=\)  \( 2^{8} \cdot 13 \cdot 17 \cdot 16843 \)
\( J_8 \)  \(=\) \(430794130940\) \(=\)  \( 2^{2} \cdot 5 \cdot 37 \cdot 9467 \cdot 61493 \)
\( J_{10} \)  \(=\) \(2176\) \(=\)  \( 2^{7} \cdot 17 \)
\( g_1 \)  \(=\) \(194465720403941544/17\)
\( g_2 \)  \(=\) \(7948719687495546/17\)
\( g_3 \)  \(=\) \(25108109106912\)

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

magma: [C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![1,-1,0],C![1,1,0]]; // simplified model
 

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: \(\Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz - 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 7xz^2 + 2z^3\) \(0\) \(12\)

2-torsion field: 4.4.4352.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 23.57577 \)
Tamagawa product: \( 3 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.491161 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 34.a
  Elliptic curve isogeny class 32.a

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 \(\Z \times \Z [\sqrt{-1}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-1}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)