Properties

Label 472.a.944.1
Conductor $472$
Discriminant $-944$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{8}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = x^5 - x^4 - 2x^3 + x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^5z - x^4z^2 - 2x^3z^3 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 3x^4 - 8x^3 + 2x^2 + 4x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -2, -1, 1]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -2, -1, 1], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 4, 2, -8, -3, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(472\) \(=\) \( 2^{3} \cdot 59 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-944\) \(=\) \( - 2^{4} \cdot 59 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(280\) \(=\)  \( 2^{3} \cdot 5 \cdot 7 \)
\( I_4 \)  \(=\) \(760\) \(=\)  \( 2^{3} \cdot 5 \cdot 19 \)
\( I_6 \)  \(=\) \(60604\) \(=\)  \( 2^{2} \cdot 109 \cdot 139 \)
\( I_{10} \)  \(=\) \(-3776\) \(=\)  \( - 2^{6} \cdot 59 \)
\( J_2 \)  \(=\) \(140\) \(=\)  \( 2^{2} \cdot 5 \cdot 7 \)
\( J_4 \)  \(=\) \(690\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 23 \)
\( J_6 \)  \(=\) \(4544\) \(=\)  \( 2^{6} \cdot 71 \)
\( J_8 \)  \(=\) \(40015\) \(=\)  \( 5 \cdot 53 \cdot 151 \)
\( J_{10} \)  \(=\) \(-944\) \(=\)  \( - 2^{4} \cdot 59 \)
\( g_1 \)  \(=\) \(-3361400000/59\)
\( g_2 \)  \(=\) \(-118335000/59\)
\( g_3 \)  \(=\) \(-5566400/59\)

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

magma: [C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]]; // minimal model
 
magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \oplus \Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((0 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(8\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((0 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(8\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - z^3\) \(0\) \(2\)
\((0 : -1 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - z^3\) \(0\) \(8\)

2-torsion field: 3.1.59.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 29.11327 \)
Tamagawa product: \( 2 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.227447 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(3\) \(4\) \(2\) \(1 + T + 2 T^{2}\)
\(59\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 12 T + 59 T^{2} )\)

Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime \(\ell\) mod-\(\ell\) image
\(2\) 2.120.3

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 \(\Q\).