Properties

Label 427.a.2989.1
Conductor 427
Discriminant -2989
Mordell-Weil group \(\Z/{14}\Z\)
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

$y^2 + (x^3 + 1)y = x^5 - x^4 - 5x^3 + 4x^2 + 4x - 4$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - x^4z^2 - 5x^3z^3 + 4x^2z^4 + 4xz^5 - 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 4x^4 - 18x^3 + 16x^2 + 16x - 15$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-4, 4, 4, -5, -1, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-4, 4, 4, -5, -1, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-15, 16, 16, -18, -4, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(427\) \(=\) \( 7 \cdot 61 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-2989\) \(=\) \( - 7^{2} \cdot 61 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(4564\) \(=\)  \( 2^{2} \cdot 7 \cdot 163 \)
\( I_4 \)  \(=\) \(-22439\) \(=\)  \( - 19 \cdot 1181 \)
\( I_6 \)  \(=\) \(-35962915\) \(=\)  \( - 5 \cdot 19 \cdot 23 \cdot 109 \cdot 151 \)
\( I_{10} \)  \(=\) \(-382592\) \(=\)  \( - 2^{7} \cdot 7^{2} \cdot 61 \)
\( J_2 \)  \(=\) \(1141\) \(=\)  \( 7 \cdot 163 \)
\( J_4 \)  \(=\) \(55180\) \(=\)  \( 2^{2} \cdot 5 \cdot 31 \cdot 89 \)
\( J_6 \)  \(=\) \(3641688\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 37 \cdot 1367 \)
\( J_8 \)  \(=\) \(277583402\) \(=\)  \( 2 \cdot 10133 \cdot 13697 \)
\( J_{10} \)  \(=\) \(-2989\) \(=\)  \( - 7^{2} \cdot 61 \)
\( g_1 \)  \(=\) \(-39466820645749/61\)
\( g_2 \)  \(=\) \(-1672794336220/61\)
\( g_3 \)  \(=\) \(-96756008472/61\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1),\, (-3 : 13 : 1)\)

magma: [C![-3,13,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

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: \(\Z/{14}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0\) \(14\)

2-torsion field: 4.2.976.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 18.61317 \)
Tamagawa product: \( 2 \)
Torsion order:\( 14 \)
Leading coefficient: \( 0.189930 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 - T + 7 T^{2} )\)
\(61\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 8 T + 61 T^{2} )\)

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 \(\Q\).