Properties

Label 3172.a.12688.1
Conductor $3172$
Discriminant $-12688$
Mordell-Weil group \(\Z \oplus \Z/{3}\Z\)
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

$y^2 + x^3y = x^2 - 2x + 1$ (homogenize, simplify)
$y^2 + x^3y = x^2z^4 - 2xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^2 - 8x + 4$ (homogenize, minimize)

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

Invariants

Conductor: \( N \)  \(=\)  \(3172\) \(=\) \( 2^{2} \cdot 13 \cdot 61 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-12688\) \(=\) \( - 2^{4} \cdot 13 \cdot 61 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(120\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(453\) \(=\)  \( 3 \cdot 151 \)
\( I_6 \)  \(=\) \(16833\) \(=\)  \( 3 \cdot 31 \cdot 181 \)
\( I_{10} \)  \(=\) \(1586\) \(=\)  \( 2 \cdot 13 \cdot 61 \)
\( J_2 \)  \(=\) \(120\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \)
\( J_4 \)  \(=\) \(298\) \(=\)  \( 2 \cdot 149 \)
\( J_6 \)  \(=\) \(-896\) \(=\)  \( - 2^{7} \cdot 7 \)
\( J_8 \)  \(=\) \(-49081\) \(=\)  \( -49081 \)
\( J_{10} \)  \(=\) \(12688\) \(=\)  \( 2^{4} \cdot 13 \cdot 61 \)
\( g_1 \)  \(=\) \(1555200000/793\)
\( g_2 \)  \(=\) \(32184000/793\)
\( g_3 \)  \(=\) \(-806400/793\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\)
\((-2 : -1 : 1)\) \((-2 : 9 : 1)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\)
\((-2 : -1 : 1)\) \((-2 : 9 : 1)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\) \((0 : -2 : 1)\) \((0 : 2 : 1)\)
\((-2 : -10 : 1)\) \((-2 : 10 : 1)\)

magma: [C![-2,-1,1],C![-2,9,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,1]]; // minimal model
 
magma: [C![-2,-10,1],C![-2,10,1],C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,-1,1],C![1,1,0],C![1,1,1]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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: \(\Z \oplus \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.052270\) \(\infty\)
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0\) \(3\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.052270\) \(\infty\)
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0\) \(3\)
Generator $D_0$ Height Order
\((0 : -2 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.052270\) \(\infty\)
\(2 \cdot(0 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 2xz^2 - 2z^3\) \(0\) \(3\)

2-torsion field: 6.0.203008.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.052270 \)
Real period: \( 19.53341 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.340337 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(3\) \(1 + 2 T + 2 T^{2}\)
\(13\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 13 T^{2} )\)
\(61\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 8 T + 61 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.10.1 no
\(3\) 3.80.1 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);