Properties

Label 10073.a.10073.1
Conductor $10073$
Discriminant $-10073$
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, 0, 2, -1]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, 0, 2, -1], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -9, 2, 9, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10073\) \(=\) \( 7 \cdot 1439 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-10073\) \(=\) \( - 7 \cdot 1439 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(500\) \(=\)  \( 2^{2} \cdot 5^{3} \)
\( I_4 \)  \(=\) \(7801\) \(=\)  \( 29 \cdot 269 \)
\( I_6 \)  \(=\) \(1091757\) \(=\)  \( 3 \cdot 17 \cdot 21407 \)
\( I_{10} \)  \(=\) \(-1289344\) \(=\)  \( - 2^{7} \cdot 7 \cdot 1439 \)
\( J_2 \)  \(=\) \(125\) \(=\)  \( 5^{3} \)
\( J_4 \)  \(=\) \(326\) \(=\)  \( 2 \cdot 163 \)
\( J_6 \)  \(=\) \(644\) \(=\)  \( 2^{2} \cdot 7 \cdot 23 \)
\( J_8 \)  \(=\) \(-6444\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 179 \)
\( J_{10} \)  \(=\) \(-10073\) \(=\)  \( - 7 \cdot 1439 \)
\( g_1 \)  \(=\) \(-30517578125/10073\)
\( g_2 \)  \(=\) \(-636718750/10073\)
\( g_3 \)  \(=\) \(-1437500/1439\)

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

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((1 : -1 : 1)\)
\((1 : -2 : 1)\) \((2 : -3 : 1)\) \((2 : -4 : 1)\) \((1 : -10 : 4)\) \((1 : -74 : 4)\)
Known points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((1 : -1 : 1)\)
\((1 : -2 : 1)\) \((2 : -3 : 1)\) \((2 : -4 : 1)\) \((1 : -10 : 4)\) \((1 : -74 : 4)\)
Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\) \((1 : -1 : 1)\)
\((1 : 1 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((1 : -64 : 4)\) \((1 : 64 : 4)\)

magma: [C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-74,4],C![1,-10,4],C![1,-2,1],C![1,-1,1],C![1,0,0],C![2,-4,1],C![2,-3,1]]; // minimal model
 
magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-64,4],C![1,64,4],C![1,-1,1],C![1,1,1],C![1,0,0],C![2,-1,1],C![2,1,1]]; // simplified model
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.163604\) \(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.132689\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -2 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.163604\) \(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.132689\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 3xz^2 + z^3\) \(0.163604\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2 - 3z^3\) \(0.132689\) \(\infty\)

2-torsion field: 5.3.161168.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.017404 \)
Real period: \( 24.24596 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.421984 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(1439\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 36 T + 1439 T^{2} )\)

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 \(\Q\).