Properties

Label 336.a.172032.1
Conductor $336$
Discriminant $-172032$
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^6 + 15x^4 - 75x^2 - 56$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 + 15x^4z^2 - 75x^2z^4 - 56z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 62x^4 - 299x^2 - 224$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-56, 0, -75, 0, 15, 0, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-56, 0, -75, 0, 15, 0, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-224, 0, -299, 0, 62, 0, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(336\) \(=\) \( 2^{4} \cdot 3 \cdot 7 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(336,2),R![1, 1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-172032\) \(=\) \( - 2^{13} \cdot 3 \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(16916\) \(=\)  \( 2^{2} \cdot 4229 \)
\( I_4 \)  \(=\) \(151117825\) \(=\)  \( 5^{2} \cdot 6044713 \)
\( I_6 \)  \(=\) \(232872423961\) \(=\)  \( 397 \cdot 586580413 \)
\( I_{10} \)  \(=\) \(-21504\) \(=\)  \( - 2^{10} \cdot 3 \cdot 7 \)
\( J_2 \)  \(=\) \(16916\) \(=\)  \( 2^{2} \cdot 4229 \)
\( J_4 \)  \(=\) \(-88822256\) \(=\)  \( - 2^{4} \cdot 103 \cdot 53897 \)
\( J_6 \)  \(=\) \(277597802496\) \(=\)  \( 2^{12} \cdot 3 \cdot 7 \cdot 3227281 \)
\( J_8 \)  \(=\) \(-798387183476800\) \(=\)  \( - 2^{6} \cdot 5^{2} \cdot 107 \cdot 991 \cdot 4705829 \)
\( J_{10} \)  \(=\) \(-172032\) \(=\)  \( - 2^{13} \cdot 3 \cdot 7 \)
\( g_1 \)  \(=\) \(-1352659309173012149/168\)
\( g_2 \)  \(=\) \(419870026410625699/168\)
\( g_3 \)  \(=\) \(-461744933079368\)

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 $\Q_{2}$.

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: \(\Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(3x^2 - 32z^2\) \(=\) \(0,\) \(6y\) \(=\) \(-35xz^2\) \(0\) \(2\)

2-torsion field: 8.4.260112384.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 0.356066 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.178033 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(13\) \(1\) \(1 + T\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 7 T^{2} )\)

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 14.a1
  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 \(\Q\).

Additional information

The Jacobian of this curve is conjectured to have the smallest conductor among all genus 2 curves whose analytic order of Sha is not a square.