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Properties

Label 800.a.8000.1
Conductor $800$
Discriminant $8000$
Mordell-Weil group \(\Z/{4}\Z\)
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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Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = -x^6 + 2x^4 + 4x^3 + 2x^2 - 1$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^6 + 2x^4z^2 + 4x^3z^3 + 2x^2z^4 - z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 2x^5 + 11x^4 + 20x^3 + 11x^2 + 2x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 0, 2, 4, 2, 0, -1]), R([1, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 0, 2, 4, 2, 0, -1], R![1, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([-3, 2, 11, 20, 11, 2, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(800\) \(=\) \( 2^{5} \cdot 5^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(8000\) \(=\) \( 2^{6} \cdot 5^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(192\) \(=\)  \( 2^{6} \cdot 3 \)
\( I_4 \)  \(=\) \(11604\) \(=\)  \( 2^{2} \cdot 3 \cdot 967 \)
\( I_6 \)  \(=\) \(322392\) \(=\)  \( 2^{3} \cdot 3 \cdot 7 \cdot 19 \cdot 101 \)
\( I_{10} \)  \(=\) \(-1000\) \(=\)  \( - 2^{3} \cdot 5^{3} \)
\( J_2 \)  \(=\) \(192\) \(=\)  \( 2^{6} \cdot 3 \)
\( J_4 \)  \(=\) \(-6200\) \(=\)  \( - 2^{3} \cdot 5^{2} \cdot 31 \)
\( J_6 \)  \(=\) \(142400\) \(=\)  \( 2^{6} \cdot 5^{2} \cdot 89 \)
\( J_8 \)  \(=\) \(-2774800\) \(=\)  \( - 2^{4} \cdot 5^{2} \cdot 7 \cdot 991 \)
\( J_{10} \)  \(=\) \(-8000\) \(=\)  \( - 2^{6} \cdot 5^{3} \)
\( g_1 \)  \(=\) \(-4076863488/125\)
\( g_2 \)  \(=\) \(27426816/5\)
\( g_3 \)  \(=\) \(-3280896/5\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model
 
magma: []; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\Q_{2}$ and $\Q_{5}$.

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: \(\Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(4\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 + z^3\) \(0\) \(4\)

2-torsion field: 4.0.320.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 5.590050 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.349378 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(6\) \(1\) \(1\)
\(5\) \(2\) \(3\) \(1\) \(( 1 - T )( 1 + T )\)

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 curve isogeny classes:
  Elliptic curve isogeny class 20.a
  Elliptic curve isogeny class 40.a

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).