Properties

Label 100325.a.100325.1
Conductor 100325
Discriminant 100325
Mordell-Weil group \(\Z\)
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

$y^2 + x^2y = x^5 - 2x^4 + 4x^3 - 4x^2 + 3x - 1$ (homogenize, simplify)
$y^2 + x^2zy = x^5z - 2x^4z^2 + 4x^3z^3 - 4x^2z^4 + 3xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 7x^4 + 16x^3 - 16x^2 + 12x - 4$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -4, 4, -2, 1], R![0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -4, 4, -2, 1]), R([0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-4, 12, -16, 16, -7, 4]))
 

Invariants

Conductor: \( N \)  =  \(100325\) = \( 5^{2} \cdot 4013 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(100325\) = \( 5^{2} \cdot 4013 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1664\) =  \( 2^{7} \cdot 13 \)
\( I_4 \)  = \(70720\) =  \( 2^{6} \cdot 5 \cdot 13 \cdot 17 \)
\( I_6 \)  = \(38080960\) =  \( 2^{6} \cdot 5 \cdot 59 \cdot 2017 \)
\( I_{10} \)  = \(410931200\) =  \( 2^{12} \cdot 5^{2} \cdot 4013 \)
\( J_2 \)  = \(208\) =  \( 2^{4} \cdot 13 \)
\( J_4 \)  = \(1066\) =  \( 2 \cdot 13 \cdot 41 \)
\( J_6 \)  = \(-2719\) =  \( - 2719 \)
\( J_8 \)  = \(-425477\) =  \( - 13 \cdot 23 \cdot 1423 \)
\( J_{10} \)  = \(100325\) =  \( 5^{2} \cdot 4013 \)
\( g_1 \)  = \(389328928768/100325\)
\( g_2 \)  = \(9592840192/100325\)
\( g_3 \)  = \(-117634816/100325\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

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: \((1 : 0 : 0),\, (2 : 3 : 1),\, (2 : -7 : 1)\)

magma: [C![1,0,0],C![2,-7,1],C![2,3,1]];
 

Number of rational Weierstrass points: \(1\)

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.396627\) \(\infty\)

2-torsion field: 5.1.1605200.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.396627 \)
Real period: \( 7.575309 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 3.004577 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(2\) \(2\) \(1\) \(1 + 2 T + 5 T^{2}\)
\(4013\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 36 T + 4013 T^{2} )\)

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 \(\Q\).