Properties

Label 847.d.847.1
Conductor 847
Discriminant -847
Mordell-Weil group \(\Z/{3}\Z\)
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

$y^2 + (x^3 + x^2 + x + 1)y = -12x^6 - 15x^5 + 9x^4 + 31x^3 + 9x^2 - 15x - 12$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -12x^6 - 15x^5z + 9x^4z^2 + 31x^3z^3 + 9x^2z^4 - 15xz^5 - 12z^6$ (dehomogenize, simplify)
$y^2 = -47x^6 - 58x^5 + 39x^4 + 128x^3 + 39x^2 - 58x - 47$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-12, -15, 9, 31, 9, -15, -12], R![1, 1, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-12, -15, 9, 31, 9, -15, -12]), R([1, 1, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-47, -58, 39, 128, 39, -58, -47]))
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-321632\) =  \( - 2^{5} \cdot 19 \cdot 23^{2} \)
\( I_4 \)  = \(6438459712\) =  \( 2^{6} \cdot 2441 \cdot 41213 \)
\( I_6 \)  = \(-518064193667072\) =  \( - 2^{18} \cdot 13 \cdot 67 \cdot 499 \cdot 4547 \)
\( I_{10} \)  = \(-3469312\) =  \( - 2^{12} \cdot 7 \cdot 11^{2} \)
\( J_2 \)  = \(-40204\) =  \( - 2^{2} \cdot 19 \cdot 23^{2} \)
\( J_4 \)  = \(281112\) =  \( 2^{3} \cdot 3 \cdot 13 \cdot 17 \cdot 53 \)
\( J_6 \)  = \(-1967560\) =  \( - 2^{3} \cdot 5 \cdot 7 \cdot 7027 \)
\( J_8 \)  = \(19956424\) =  \( 2^{3} \cdot 2494553 \)
\( J_{10} \)  = \(-847\) =  \( - 7 \cdot 11^{2} \)
\( g_1 \)  = \(105037970421355597057024/847\)
\( g_2 \)  = \(18267839107785466368/847\)
\( g_3 \)  = \(454326923025280/121\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable except over $\R$.

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: \(\Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(18x^2 + 29xz + 18z^2\) \(=\) \(0,\) \(162y\) \(=\) \(-77xz^2 - 126z^3\) \(0\) \(3\)

2-torsion field: 6.0.54208.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 1.179534 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.262118 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

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

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 77.b1
  Elliptic curve 11.a1

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) 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 \(\Q\).