Properties

Label 847.b.9317.1
Conductor 847
Discriminant 9317
Mordell-Weil group \(\Z/{10}\Z\)
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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 2, -3, 2, 1]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 2, -3, 2, 1], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -4, 10, -12, 9, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(847\) \(=\) \( 7 \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(9317\) \(=\) \( 7 \cdot 11^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-1216\) \(=\)  \( - 2^{6} \cdot 19 \)
\( I_4 \)  \(=\) \(94912\) \(=\)  \( 2^{6} \cdot 1483 \)
\( I_6 \)  \(=\) \(-28957760\) \(=\)  \( - 2^{6} \cdot 5 \cdot 13 \cdot 6961 \)
\( I_{10} \)  \(=\) \(38162432\) \(=\)  \( 2^{12} \cdot 7 \cdot 11^{3} \)
\( J_2 \)  \(=\) \(-152\) \(=\)  \( - 2^{3} \cdot 19 \)
\( J_4 \)  \(=\) \(-26\) \(=\)  \( - 2 \cdot 13 \)
\( J_6 \)  \(=\) \(401\) \(=\)  \( 401 \)
\( J_8 \)  \(=\) \(-15407\) \(=\)  \( - 7 \cdot 31 \cdot 71 \)
\( J_{10} \)  \(=\) \(9317\) \(=\)  \( 7 \cdot 11^{3} \)
\( g_1 \)  \(=\) \(-81136812032/9317\)
\( g_2 \)  \(=\) \(91307008/9317\)
\( g_3 \)  \(=\) \(9264704/9317\)

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

magma: [C![0,-1,1],C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(1\)

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: \(\Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(3xz^2 - 2z^3\) \(0\) \(10\)

2-torsion field: 6.0.664048.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 16.82727 \)
Tamagawa product: \( 2 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.336545 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 7 T^{2} )\)
\(11\) \(2\) \(3\) \(2\) \(( 1 - T )^{2}\)

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 77.c2
  Elliptic curve 11.a3

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 \(\Q\).