Properties

Label 27225.a.81675.1
Conductor $27225$
Discriminant $-81675$
Mordell-Weil group \(\Z \times \Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = 2x^4 + 2x^3 + 5x^2 + 3x + 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 2x^4z^2 + 2x^3z^3 + 5x^2z^4 + 3xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^4 + 10x^3 + 20x^2 + 12x + 9$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 3, 5, 2, 2]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 3, 5, 2, 2], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, 12, 20, 10, 8, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(27225\) \(=\) \( 3^{2} \cdot 5^{2} \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-81675\) \(=\) \( - 3^{3} \cdot 5^{2} \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2060\) \(=\)  \( 2^{2} \cdot 5 \cdot 103 \)
\( I_4 \)  \(=\) \(56521\) \(=\)  \( 29 \cdot 1949 \)
\( I_6 \)  \(=\) \(34621947\) \(=\)  \( 3^{2} \cdot 31^{2} \cdot 4003 \)
\( I_{10} \)  \(=\) \(10454400\) \(=\)  \( 2^{7} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2} \)
\( J_2 \)  \(=\) \(515\) \(=\)  \( 5 \cdot 103 \)
\( J_4 \)  \(=\) \(8696\) \(=\)  \( 2^{3} \cdot 1087 \)
\( J_6 \)  \(=\) \(172224\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 13 \cdot 23 \)
\( J_8 \)  \(=\) \(3268736\) \(=\)  \( 2^{7} \cdot 25537 \)
\( J_{10} \)  \(=\) \(81675\) \(=\)  \( 3^{3} \cdot 5^{2} \cdot 11^{2} \)
\( g_1 \)  \(=\) \(1449092592875/3267\)
\( g_2 \)  \(=\) \(47511769960/3267\)
\( g_3 \)  \(=\) \(203013824/363\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((0 : -2 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((7 : 1321 : 8)\) \((7 : -2176 : 8)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((0 : -2 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((7 : 1321 : 8)\) \((7 : -2176 : 8)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -3 : 1)\) \((0 : 3 : 1)\) \((-1 : -4 : 1)\) \((-1 : 4 : 1)\)
\((7 : -3497 : 8)\) \((7 : 3497 : 8)\)

magma: [C![-1,-2,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0],C![7,-2176,8],C![7,1321,8]]; // minimal model
 
magma: [C![-1,-4,1],C![-1,4,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0],C![7,-3497,8],C![7,3497,8]]; // simplified model
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + 2xz + 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(xz^2 - z^3\) \(0.393627\) \(\infty\)
\((0 : -2 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.365989\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + 2xz + 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(xz^2 - z^3\) \(0.393627\) \(\infty\)
\((0 : -2 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.365989\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 + 2xz + 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 + 2xz^2 - z^3\) \(0.393627\) \(\infty\)
\((0 : -3 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3z^3\) \(0.365989\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0\) \(2\)

2-torsion field: 8.0.741200625.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.143872 \)
Real period: \( 9.543118 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.686496 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(2\) \(3\) \(2\) \(( 1 + T )^{2}\)
\(5\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)
\(11\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).