Properties

Label 1575.a.165375.1
Conductor 1575
Discriminant -165375
Mordell-Weil group \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)
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

$y^2 + (x^2 + x + 1)y = -x^5 + 2x^4 + x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = -x^5z + 2x^4z^2 + x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = -4x^5 + 9x^4 + 2x^3 + 7x^2 - 6x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 1, 0, 2, -1]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 1, 0, 2, -1], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, -6, 7, 2, 9, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1575\) \(=\) \( 3^{2} \cdot 5^{2} \cdot 7 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-165375\) \(=\) \( - 3^{3} \cdot 5^{3} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-24\) \(=\)  \( - 2^{3} \cdot 3 \)
\( I_4 \)  \(=\) \(77220\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \cdot 13 \)
\( I_6 \)  \(=\) \(-20329560\) \(=\)  \( - 2^{3} \cdot 3^{2} \cdot 5 \cdot 149 \cdot 379 \)
\( I_{10} \)  \(=\) \(-677376000\) \(=\)  \( - 2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(-3\) \(=\)  \( -3 \)
\( J_4 \)  \(=\) \(-804\) \(=\)  \( - 2^{2} \cdot 3 \cdot 67 \)
\( J_6 \)  \(=\) \(34624\) \(=\)  \( 2^{6} \cdot 541 \)
\( J_8 \)  \(=\) \(-187572\) \(=\)  \( - 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 \cdot 29 \)
\( J_{10} \)  \(=\) \(-165375\) \(=\)  \( - 3^{3} \cdot 5^{3} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(9/6125\)
\( g_2 \)  \(=\) \(-804/6125\)
\( g_3 \)  \(=\) \(-34624/18375\)

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

All points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((2 : 0 : 1)\) \((-1 : 2 : 1)\)
\((-1 : -3 : 1)\) \((1 : -3 : 1)\) \((2 : -7 : 1)\) \((1 : -42 : 4)\)

magma: [C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-42,4],C![1,-3,1],C![1,0,0],C![1,0,1],C![2,-7,1],C![2,0,1]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z \times \Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -3 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2\) \(0.027182\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\((1 : -42 : 4) - (1 : 0 : 0)\) \(4x - z\) \(=\) \(0,\) \(32y\) \(=\) \(-21z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(3\)
Regulator: \( 0.027182 \)
Real period: \( 17.77977 \)
Tamagawa product: \( 8 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.241645 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(2\) \(3\) \(2\) \(1 + 2 T + 3 T^{2}\)
\(5\) \(2\) \(3\) \(2\) \(1 + 2 T + 5 T^{2}\)
\(7\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 7 T^{2} )\)

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 \(\Q\).