Properties

Label 1408.a.180224.1
Conductor 1408
Discriminant 180224
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = x^4 - x^3 - 5x + 1$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^4z^2 - x^3z^3 - 5xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 7x^4 + 3x^2 - 18x + 5$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -5, 0, -1, 1]), R([1, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -5, 0, -1, 1], R![1, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([5, -18, 3, 0, 7, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1408\) \(=\) \( 2^{7} \cdot 11 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1408,2),R![1, 0, 2]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(180224\) \(=\) \( 2^{14} \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-2976\) \(=\)  \( - 2^{5} \cdot 3 \cdot 31 \)
\( I_4 \)  \(=\) \(761856\) \(=\)  \( 2^{13} \cdot 3 \cdot 31 \)
\( I_6 \)  \(=\) \(-531038208\) \(=\)  \( - 2^{16} \cdot 3 \cdot 37 \cdot 73 \)
\( I_{10} \)  \(=\) \(738197504\) \(=\)  \( 2^{26} \cdot 11 \)
\( J_2 \)  \(=\) \(-372\) \(=\)  \( - 2^{2} \cdot 3 \cdot 31 \)
\( J_4 \)  \(=\) \(-2170\) \(=\)  \( - 2 \cdot 5 \cdot 7 \cdot 31 \)
\( J_6 \)  \(=\) \(-17276\) \(=\)  \( - 2^{2} \cdot 7 \cdot 617 \)
\( J_8 \)  \(=\) \(429443\) \(=\)  \( 7 \cdot 31 \cdot 1979 \)
\( J_{10} \)  \(=\) \(180224\) \(=\)  \( 2^{14} \cdot 11 \)
\( g_1 \)  \(=\) \(-6956883693/176\)
\( g_2 \)  \(=\) \(872727345/1408\)
\( g_3 \)  \(=\) \(-37355031/2816\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -2 : 1)\)

magma: [C![1,-2,1],C![1,-1,0],C![1,0,0]];
 

Number of rational Weierstrass points: \(1\)

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: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0\) \(6\)

2-torsion field: 6.0.30976.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 9.349938 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.519441 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(7\) \(14\) \(2\) \(1 + 2 T^{2}\)
\(11\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 11 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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