Properties

Label 15256.a.122048.1
Conductor 15256
Discriminant 122048
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

$y^2 + (x + 1)y = x^6 - x^4 + 2x^2 + x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^6 - x^4z^2 + 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 4x^4 + 9x^2 + 6x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 2, 0, -1, 0, 1], R![1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 2, 0, -1, 0, 1]), R([1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 6, 9, 0, -4, 0, 4]))
 

Invariants

Conductor: \( N \)  =  \(15256\) = \( 2^{3} \cdot 1907 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(122048\) = \( 2^{6} \cdot 1907 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-384\) =  \( - 2^{7} \cdot 3 \)
\( I_4 \)  = \(67776\) =  \( 2^{6} \cdot 3 \cdot 353 \)
\( I_6 \)  = \(-13663488\) =  \( - 2^{8} \cdot 3 \cdot 17791 \)
\( I_{10} \)  = \(499908608\) =  \( 2^{18} \cdot 1907 \)
\( J_2 \)  = \(-48\) =  \( - 2^{4} \cdot 3 \)
\( J_4 \)  = \(-610\) =  \( - 2 \cdot 5 \cdot 61 \)
\( J_6 \)  = \(14052\) =  \( 2^{2} \cdot 3 \cdot 1171 \)
\( J_8 \)  = \(-261649\) =  \( - 19 \cdot 47 \cdot 293 \)
\( J_{10} \)  = \(122048\) =  \( 2^{6} \cdot 1907 \)
\( g_1 \)  = \(-3981312/1907\)
\( g_2 \)  = \(1054080/1907\)
\( g_3 \)  = \(505872/1907\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((-1 : -1 : 2)\) \((1 : -3 : 1)\) \((-1 : -3 : 2)\) \((-1 : -17 : 4)\) \((2 : 17 : 3)\)
\((-1 : -31 : 4)\) \((2 : -62 : 3)\)

magma: [C![-1,-31,4],C![-1,-17,4],C![-1,-3,2],C![-1,-1,1],C![-1,-1,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,1,1],C![2,-62,3],C![2,17,3]];
 

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
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.264086\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0.021732\) \(\infty\)

2-torsion field: 6.2.488192.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.005663 \)
Real period: \( 17.79322 \)
Tamagawa product: \( 6 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.604627 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(6\) \(3\) \(6\) \(1 + T\)
\(1907\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 29 T + 1907 T^{2} )\)

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 \(\Q\).