Properties

Label 816.b.52224.1
Conductor $816$
Discriminant $-52224$
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^6 - 12x^4 - 27x^2 - 17$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 - 12x^4z^2 - 27x^2z^4 - 17z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 46x^4 - 107x^2 - 68$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-17, 0, -27, 0, -12, 0, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-17, 0, -27, 0, -12, 0, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-68, 0, -107, 0, -46, 0, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(816\) \(=\) \( 2^{4} \cdot 3 \cdot 17 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-52224\) \(=\) \( - 2^{10} \cdot 3 \cdot 17 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(15964\) \(=\)  \( 2^{2} \cdot 13 \cdot 307 \)
\( I_4 \)  \(=\) \(2380825\) \(=\)  \( 5^{2} \cdot 95233 \)
\( I_6 \)  \(=\) \(11444690699\) \(=\)  \( 13 \cdot 13523 \cdot 65101 \)
\( I_{10} \)  \(=\) \(6528\) \(=\)  \( 2^{7} \cdot 3 \cdot 17 \)
\( J_2 \)  \(=\) \(15964\) \(=\)  \( 2^{2} \cdot 13 \cdot 307 \)
\( J_4 \)  \(=\) \(9031504\) \(=\)  \( 2^{4} \cdot 163 \cdot 3463 \)
\( J_6 \)  \(=\) \(6282991104\) \(=\)  \( 2^{9} \cdot 3 \cdot 13 \cdot 17 \cdot 83 \cdot 223 \)
\( J_8 \)  \(=\) \(4683401370560\) \(=\)  \( 2^{6} \cdot 5 \cdot 11 \cdot 139 \cdot 9572027 \)
\( J_{10} \)  \(=\) \(52224\) \(=\)  \( 2^{10} \cdot 3 \cdot 17 \)
\( g_1 \)  \(=\) \(1012531723491160951/51\)
\( g_2 \)  \(=\) \(35882713644370099/51\)
\( g_3 \)  \(=\) \(30660536527816\)

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

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable except over $\R$.

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
\(D_0 - D_\infty\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)

2-torsion field: 8.0.95883264.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 2.423742 \)
Tamagawa product: \( 3 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.403957 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(10\) \(3\) \(1 - T\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(17\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 17 T^{2} )\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 34.a3
  Elliptic curve 24.a1

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 \(\Q\).