Properties

Label 7225.a.36125.1
Conductor 7225
Discriminant -36125
Mordell-Weil group \(\Z \times \Z \times \Z/{2}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -3, 0, 2, -1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -3, 0, 2, -1]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -4, -12, 2, 8, -4, 1]))
 

Invariants

Conductor: \( N \)  =  \(7225\) = \( 5^{2} \cdot 17^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-36125\) = \( - 5^{3} \cdot 17^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1960\) =  \( 2^{3} \cdot 5 \cdot 7^{2} \)
\( I_4 \)  = \(-8156\) =  \( - 2^{2} \cdot 2039 \)
\( I_6 \)  = \(-20952536\) =  \( - 2^{3} \cdot 11 \cdot 457 \cdot 521 \)
\( I_{10} \)  = \(-147968000\) =  \( - 2^{12} \cdot 5^{3} \cdot 17^{2} \)
\( J_2 \)  = \(245\) =  \( 5 \cdot 7^{2} \)
\( J_4 \)  = \(2586\) =  \( 2 \cdot 3 \cdot 431 \)
\( J_6 \)  = \(64636\) =  \( 2^{2} \cdot 11 \cdot 13 \cdot 113 \)
\( J_8 \)  = \(2287106\) =  \( 2 \cdot 19 \cdot 139 \cdot 433 \)
\( J_{10} \)  = \(-36125\) =  \( - 5^{3} \cdot 17^{2} \)
\( g_1 \)  = \(-7061881225/289\)
\( g_2 \)  = \(-304240314/289\)
\( g_3 \)  = \(-155191036/1445\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-1 : -1 : 2)\) \((2 : -2 : 1)\) \((-1 : -6 : 2)\) \((2 : -7 : 1)\) \((5 : -57 : 3)\) \((5 : -95 : 3)\)

magma: [C![-1,-6,2],C![-1,-1,1],C![-1,-1,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0],C![2,-7,1],C![2,-2,1],C![5,-95,3],C![5,-57,3]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -6 : 2) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (2x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - 2z^3\) \(0.289355\) \(\infty\)
\((-1 : 1 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - 2z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.146893\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)

2-torsion field: 8.0.46240000.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.037430 \)
Real period: \( 18.34560 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.343342 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(3\) \(2\) \(2\) \(( 1 + T )^{2}\)
\(17\) \(2\) \(2\) \(1\) \(( 1 + T )^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

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 \(\Q\).