Properties

Label 847.d.456533.1
Conductor 847
Discriminant 456533
Mordell-Weil group \(\Z/{15}\Z\)
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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, -24, 37, 3, -22, -9, -1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, -24, 37, 3, -22, -9, -1]), R([1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, -24, 37, 3, -22, -9, -1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([17, -96, 148, 12, -88, -36, -4]))
 

$y^2 + y = -x^6 - 9x^5 - 22x^4 + 3x^3 + 37x^2 - 24x + 4$ (homogenize, simplify)
$y^2 + z^3y = -x^6 - 9x^5z - 22x^4z^2 + 3x^3z^3 + 37x^2z^4 - 24xz^5 + 4z^6$ (dehomogenize, simplify)
$y^2 = -4x^6 - 36x^5 - 88x^4 + 12x^3 + 148x^2 - 96x + 17$ (minimize, homogenize)

Invariants

\( N \)  =  \(847\) = \( 7 \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(456533\) = \( 7^{3} \cdot 11^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(363808\) =  \( 2^{5} \cdot 11369 \)
\( I_4 \)  = \(162112\) =  \( 2^{6} \cdot 17 \cdot 149 \)
\( I_6 \)  = \(19446212608\) =  \( 2^{11} \cdot 9495221 \)
\( I_{10} \)  = \(1869959168\) =  \( 2^{12} \cdot 7^{3} \cdot 11^{3} \)
\( J_2 \)  = \(45476\) =  \( 2^{2} \cdot 11369 \)
\( J_4 \)  = \(86167752\) =  \( 2^{3} \cdot 3 \cdot 11 \cdot 23^{2} \cdot 617 \)
\( J_6 \)  = \(217689875480\) =  \( 2^{3} \cdot 5 \cdot 7^{3} \cdot 11^{2} \cdot 131129 \)
\( J_8 \)  = \(618695823148744\) =  \( 2^{3} \cdot 11^{2} \cdot 571 \cdot 1119349523 \)
\( J_{10} \)  = \(456533\) =  \( 7^{3} \cdot 11^{3} \)
\( g_1 \)  = \(194496275421254111077376/456533\)
\( g_2 \)  = \(736713878289412204032/41503\)
\( g_3 \)  = \(10847340081772160/11\)

Automorphism group

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

Rational points

magma: [];
 

This curve has no rational points.

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

Number of rational Weierstrass points: \(0\)

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);
 

This curve is locally solvable except over $\Q_{11}$.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{15}\Z\)

Generator Height Order
\(2x^2 + 5xz - 4z^2\) \(=\) \(0,\) \(4y\) \(=\) \(xz^2 - 4z^3\) \(0\) \(15\)

2-torsion field: 6.0.54208.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 9.829455 \)
Tamagawa product: \( 3 \)
Torsion order:\( 15 \)
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
\(7\) \(3\) \(1\) \(3\) \(( 1 - T )( 1 + 2 T + 7 T^{2} )\)
\(11\) \(3\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

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.b2
  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 \(2\) 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 \(\Q\).