Properties

Label 1272.a.122112.1
Conductor 1272
Discriminant -122112
Mordell-Weil group \(\Z/{2}\Z \times \Z/{6}\Z\)
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

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 2, 4, 3], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -4, -2, 8, 17, 12]))
 

$y^2 + (x^2 + 1)y = 3x^5 + 4x^4 + 2x^3 - x^2 - x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = 3x^5z + 4x^4z^2 + 2x^3z^3 - x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 12x^5 + 17x^4 + 8x^3 - 2x^2 - 4x + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(1272\) = \( 2^{3} \cdot 3 \cdot 53 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-122112\) = \( - 2^{8} \cdot 3^{2} \cdot 53 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(-992\) =  \( - 2^{5} \cdot 31 \)
\( I_4 \)  = \(-321728\) =  \( - 2^{6} \cdot 11 \cdot 457 \)
\( I_6 \)  = \(18153984\) =  \( 2^{9} \cdot 3 \cdot 53 \cdot 223 \)
\( I_{10} \)  = \(-500170752\) =  \( - 2^{20} \cdot 3^{2} \cdot 53 \)
\( J_2 \)  = \(-124\) =  \( - 2^{2} \cdot 31 \)
\( J_4 \)  = \(3992\) =  \( 2^{3} \cdot 499 \)
\( J_6 \)  = \(79504\) =  \( 2^{4} \cdot 4969 \)
\( J_8 \)  = \(-6448640\) =  \( - 2^{9} \cdot 5 \cdot 11 \cdot 229 \)
\( J_{10} \)  = \(-122112\) =  \( - 2^{8} \cdot 3^{2} \cdot 53 \)
\( g_1 \)  = \(114516604/477\)
\( g_2 \)  = \(29731418/477\)
\( g_3 \)  = \(-4775209/477\)

Automorphism group

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

Rational points

magma: [C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,-15,3],C![1,0,0]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (1 : -15 : 3)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(3\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{2}\Z \times \Z/{6}\Z\)

Generator Height Order
\(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(6\)

2-torsion field: 3.1.212.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 14.39791 \)
Tamagawa product: \( 4 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.399942 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(8\) \(3\) \(2\) \(1 + T\)
\(3\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 - T + 3 T^{2} )\)
\(53\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 6 T + 53 T^{2} )\)

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 \(\Q\).