Properties

Label 389.a.389.1
Conductor 389
Discriminant 389
Mordell-Weil group \(\Z/{10}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = x^5 - 2x^4 - 8x^3 + 16x + 7$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = x^5z - 2x^4z^2 - 8x^3z^3 + 16xz^5 + 7z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 6x^4 - 32x^3 + x^2 + 64x + 28$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([7, 16, 0, -8, -2, 1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![7, 16, 0, -8, -2, 1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([28, 64, 1, -32, -6, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(389\) \(=\) \( 389 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(389\) \(=\) \( 389 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(9760\) \(=\)  \( 2^{5} \cdot 5 \cdot 61 \)
\( I_4 \)  \(=\) \(817600\) \(=\)  \( 2^{6} \cdot 5^{2} \cdot 7 \cdot 73 \)
\( I_6 \)  \(=\) \(2882646464\) \(=\)  \( 2^{6} \cdot 45041351 \)
\( I_{10} \)  \(=\) \(1593344\) \(=\)  \( 2^{12} \cdot 389 \)
\( J_2 \)  \(=\) \(1220\) \(=\)  \( 2^{2} \cdot 5 \cdot 61 \)
\( J_4 \)  \(=\) \(53500\) \(=\)  \( 2^{2} \cdot 5^{3} \cdot 107 \)
\( J_6 \)  \(=\) \(2084961\) \(=\)  \( 3 \cdot 694987 \)
\( J_8 \)  \(=\) \(-79649395\) \(=\)  \( - 5 \cdot 7 \cdot 2275697 \)
\( J_{10} \)  \(=\) \(389\) \(=\)  \( 389 \)
\( g_1 \)  \(=\) \(2702708163200000/389\)
\( g_2 \)  \(=\) \(97147868000000/389\)
\( g_3 \)  \(=\) \(3103255952400/389\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 5 : 1),\, (2 : -5 : 1)\)

magma: [C![-2,5,1],C![1,-1,0],C![1,0,0],C![2,-5,1]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 - 9z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-6xz^2 - z^3\) \(0\) \(10\)

2-torsion field: 4.4.6224.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 19.79862 \)
Tamagawa product: \( 1 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.197986 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(389\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 10 T + 389 T^{2} )\)

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 \(\Q\).