Properties

Label 1083.b.390963.1
Conductor 1083
Discriminant -390963
Mordell-Weil group trivial
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = -x^6 + 3x^5 - 50x^4 + 95x^3 - 14x^2 - 33x - 6$ (homogenize, simplify)
$y^2 + z^3y = -x^6 + 3x^5z - 50x^4z^2 + 95x^3z^3 - 14x^2z^4 - 33xz^5 - 6z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 + 12x^5 - 200x^4 + 380x^3 - 56x^2 - 132x - 23$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, -33, -14, 95, -50, 3, -1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, -33, -14, 95, -50, 3, -1], R![1]);
 
sage: X = HyperellipticCurve(R([-23, -132, -56, 380, -200, 12, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(1083\) = \( 3 \cdot 19^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-390963\) = \( - 3 \cdot 19^{4} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(601760\) =  \( 2^{5} \cdot 5 \cdot 3761 \)
\( I_4 \)  = \(31128254272\) =  \( 2^{6} \cdot 486378973 \)
\( I_6 \)  = \(4413239365215232\) =  \( 2^{10} \cdot 17 \cdot 97 \cdot 2857 \cdot 914801 \)
\( I_{10} \)  = \(-1601384448\) =  \( - 2^{12} \cdot 3 \cdot 19^{4} \)
\( J_2 \)  = \(75220\) =  \( 2^{2} \cdot 5 \cdot 3761 \)
\( J_4 \)  = \(-88500632\) =  \( - 2^{3} \cdot 11 \cdot 19 \cdot 41 \cdot 1291 \)
\( J_6 \)  = \(98386538568\) =  \( 2^{3} \cdot 3 \cdot 19^{3} \cdot 597673 \)
\( J_8 \)  = \(-107931608328616\) =  \( - 2^{3} \cdot 19^{2} \cdot 37 \cdot 1010065961 \)
\( J_{10} \)  = \(-390963\) =  \( - 3 \cdot 19^{4} \)
\( g_1 \)  = \(-2408056349828975363200000/390963\)
\( g_2 \)  = \(1982406707133537344000/20577\)
\( g_3 \)  = \(-27053302090985600/19\)

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 6.0.69312.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 0.132918 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.265837 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(19\) \(4\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

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 19.a1
  Elliptic curve 57.b1

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 \(\Q\).