Properties

Label 1125.a.151875.1
Conductor 1125
Discriminant -151875
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = 15x^5 + 50x^4 + 55x^3 + 22x^2 + 3x$ (homogenize, simplify)
$y^2 + xz^2y = 15x^5z + 50x^4z^2 + 55x^3z^3 + 22x^2z^4 + 3xz^5$ (dehomogenize, simplify)
$y^2 = 60x^5 + 200x^4 + 220x^3 + 89x^2 + 12x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 22, 55, 50, 15]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, 22, 55, 50, 15], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, 12, 89, 220, 200, 60]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(1125\) = \( 3^{2} \cdot 5^{3} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-151875\) = \( - 3^{5} \cdot 5^{4} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(34400\) =  \( 2^{5} \cdot 5^{2} \cdot 43 \)
\( I_4 \)  = \(9793600\) =  \( 2^{6} \cdot 5^{2} \cdot 6121 \)
\( I_6 \)  = \(99603070400\) =  \( 2^{6} \cdot 5^{2} \cdot 62251919 \)
\( I_{10} \)  = \(-622080000\) =  \( - 2^{12} \cdot 3^{5} \cdot 5^{4} \)
\( J_2 \)  = \(4300\) =  \( 2^{2} \cdot 5^{2} \cdot 43 \)
\( J_4 \)  = \(668400\) =  \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 557 \)
\( J_6 \)  = \(132975225\) =  \( 3^{2} \cdot 5^{2} \cdot 37 \cdot 15973 \)
\( J_8 \)  = \(31258726875\) =  \( 3^{3} \cdot 5^{4} \cdot 211 \cdot 8779 \)
\( J_{10} \)  = \(-151875\) =  \( - 3^{5} \cdot 5^{4} \)
\( g_1 \)  = \(-2352135088000000/243\)
\( g_2 \)  = \(-28342655360000/81\)
\( g_3 \)  = \(-437104339600/27\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 18 : 3)\)

magma: [C![-4,18,3],C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(3\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-4 : 18 : 3) - (1 : 0 : 0)\) \(3x + 4z\) \(=\) \(0,\) \(3y\) \(=\) \(2z^3\) \(0\) \(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 3.1.300.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 1.964401 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.491100 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(5\) \(2\) \(2\) \(( 1 + T )^{2}\)
\(5\) \(4\) \(3\) \(2\) \(1 - T\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 15.a2
  Elliptic curve 75.c1

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(3\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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