Properties

Label 100121.b.700847.1
Conductor 100121
Discriminant 700847
Mordell-Weil group \(\Z \times \Z\)
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

$y^2 + (x^3 + x + 1)y = -x^4 + 3x^2 + 6x + 4$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + 3x^2z^4 + 6xz^5 + 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 + 2x^3 + 13x^2 + 26x + 17$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 6, 3, 0, -1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 6, 3, 0, -1]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([17, 26, 13, 2, -2, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(100121\) = \( 7 \cdot 14303 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(700847\) = \( 7^{2} \cdot 14303 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-3640\) =  \( - 2^{3} \cdot 5 \cdot 7 \cdot 13 \)
\( I_4 \)  = \(284836\) =  \( 2^{2} \cdot 71209 \)
\( I_6 \)  = \(-342400472\) =  \( - 2^{3} \cdot 42800059 \)
\( I_{10} \)  = \(2870669312\) =  \( 2^{12} \cdot 7^{2} \cdot 14303 \)
\( J_2 \)  = \(-455\) =  \( - 5 \cdot 7 \cdot 13 \)
\( J_4 \)  = \(5659\) =  \( 5659 \)
\( J_6 \)  = \(1397\) =  \( 11 \cdot 127 \)
\( J_8 \)  = \(-8164979\) =  \( - 29 \cdot 281551 \)
\( J_{10} \)  = \(700847\) =  \( 7^{2} \cdot 14303 \)
\( g_1 \)  = \(-397979684375/14303\)
\( g_2 \)  = \(-10878720125/14303\)
\( g_3 \)  = \(5902325/14303\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((2 : 1 : 1)\) \((-1 : 9 : 2)\)
\((-1 : -12 : 2)\) \((2 : -12 : 1)\) \((-4 : 36 : 3)\) \((-4 : 37 : 3)\)

magma: [C![-4,36,3],C![-4,37,3],C![-1,-12,2],C![-1,0,1],C![-1,1,1],C![-1,9,2],C![1,-1,0],C![1,0,0],C![2,-12,1],C![2,1,1]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 1 : 1) + (-1 : 9 : 2) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x + z) (2x + z)\) \(=\) \(0,\) \(4y\) \(=\) \(xz^2 + 5z^3\) \(0.487205\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 3z^3\) \(0.069459\) \(\infty\)

2-torsion field: 6.2.915392.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.033091 \)
Real period: \( 14.43539 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.955378 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(7\) \(2\) \(1\) \(2\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(14303\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 170 T + 14303 T^{2} )\)

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 \(\Q\).