Properties

Label 1197.a.410571.1
Conductor 1197
Discriminant -410571
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

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -3, -7, 12, 1], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -3, -7, 12, 1]), R([1, 0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -3, -7, 12, 1], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 4, -10, -28, 49, 4]))
 

$y^2 + (x^2 + 1)y = x^5 + 12x^4 - 7x^3 - 3x^2 + x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^5z + 12x^4z^2 - 7x^3z^3 - 3x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 49x^4 - 28x^3 - 10x^2 + 4x + 1$ (minimize, homogenize)

Invariants

\( N \)  =  \(1197\) = \( 3^{2} \cdot 7 \cdot 19 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-410571\) = \( - 3^{2} \cdot 7^{4} \cdot 19 \)
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);
 

\( I_2 \)  = \(13184\) =  \( 2^{7} \cdot 103 \)
\( I_4 \)  = \(11308480\) =  \( 2^{6} \cdot 5 \cdot 35339 \)
\( I_6 \)  = \(37014592960\) =  \( 2^{6} \cdot 5 \cdot 8699 \cdot 13297 \)
\( I_{10} \)  = \(-1681698816\) =  \( - 2^{12} \cdot 3^{2} \cdot 7^{4} \cdot 19 \)
\( J_2 \)  = \(1648\) =  \( 2^{4} \cdot 103 \)
\( J_4 \)  = \(-4634\) =  \( - 2 \cdot 7 \cdot 331 \)
\( J_6 \)  = \(23921\) =  \( 19 \cdot 1259 \)
\( J_8 \)  = \(4486963\) =  \( 13 \cdot 17 \cdot 79 \cdot 257 \)
\( J_{10} \)  = \(-410571\) =  \( - 3^{2} \cdot 7^{4} \cdot 19 \)
\( g_1 \)  = \(-12155869717331968/410571\)
\( g_2 \)  = \(2962986082304/58653\)
\( g_3 \)  = \(-3419323136/21609\)

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,-34,4],C![0,-1,1],C![0,0,1],C![1,0,0]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (0 : -1 : 1),\, (-1 : -34 : 4)\)

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

Number of rational Weierstrass points: \(2\)

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

Generator Height Order
\(x (4x + z)\) \(=\) \(0,\) \(8y\) \(=\) \(17xz^2\) \(0\) \(10\)

2-torsion field: 4.2.2736.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 18.77804 \)
Tamagawa product: \( 2 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.375560 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(2\) \(2\) \(1\) \(1 + T^{2}\)
\(7\) \(4\) \(1\) \(2\) \(( 1 + T )( 1 - 3 T + 7 T^{2} )\)
\(19\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 19 T^{2} )\)

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 \(\Q\).