Properties

Label 882.a.302526.1
Conductor 882
Discriminant -302526
Mordell-Weil group \(\Z/{2}\Z \times \Z/{4}\Z\)
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

$y^2 + (x^3 + 1)y = x^5 - 2x^4 - 5x^3 + 11x^2 - 12x + 5$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 2x^4z^2 - 5x^3z^3 + 11x^2z^4 - 12xz^5 + 5z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 8x^4 - 18x^3 + 44x^2 - 48x + 21$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, -12, 11, -5, -2, 1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, -12, 11, -5, -2, 1]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([21, -48, 44, -18, -8, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(882\) = \( 2 \cdot 3^{2} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-302526\) = \( - 2 \cdot 3^{2} \cdot 7^{5} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-5144\) =  \( - 2^{3} \cdot 643 \)
\( I_4 \)  = \(-1133564\) =  \( - 2^{2} \cdot 53 \cdot 5347 \)
\( I_6 \)  = \(-1323715192\) =  \( - 2^{3} \cdot 165464399 \)
\( I_{10} \)  = \(-1239146496\) =  \( - 2^{13} \cdot 3^{2} \cdot 7^{5} \)
\( J_2 \)  = \(-643\) =  \( - 643 \)
\( J_4 \)  = \(29035\) =  \( 5 \cdot 5807 \)
\( J_6 \)  = \(3791761\) =  \( 151 \cdot 25111 \)
\( J_8 \)  = \(-820283387\) =  \( - 7 \cdot 11 \cdot 1367 \cdot 7793 \)
\( J_{10} \)  = \(-302526\) =  \( - 2 \cdot 3^{2} \cdot 7^{5} \)
\( g_1 \)  = \(109914468611443/302526\)
\( g_2 \)  = \(7718888172745/302526\)
\( g_3 \)  = \(-1567699793689/302526\)

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

All points: \((1 : 0 : 0),\, (1 : -1 : 0)\)

magma: [C![1,-1,0],C![1,0,0]];
 

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: \(\Z/{2}\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + 6xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-13xz^2 + 2z^3\) \(0\) \(4\)

2-torsion field: 8.0.796594176.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 12.54262 \)
Tamagawa product: \( 2 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.391956 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 21.a5
  Elliptic curve 42.a4

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(4\) 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 \(\Q\).