Properties

Label 847.a.847.1
Conductor 847
Discriminant -847
Mordell-Weil group \(\Z \times \Z/{5}\Z\)
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

$y^2 + (x^3 + x^2 + x + 1)y = x^4 + x^3 + x^2$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = x^4z^2 + x^3z^3 + x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 7x^4 + 8x^3 + 7x^2 + 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 1, 1], R![1, 1, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 1, 1]), R([1, 1, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 2, 7, 8, 7, 2, 1]))
 

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

\( I_2 \)  = \(-480\) =  \( - 2^{5} \cdot 3 \cdot 5 \)
\( I_4 \)  = \(4416\) =  \( 2^{6} \cdot 3 \cdot 23 \)
\( I_6 \)  = \(-439296\) =  \( - 2^{10} \cdot 3 \cdot 11 \cdot 13 \)
\( I_{10} \)  = \(-3469312\) =  \( - 2^{12} \cdot 7 \cdot 11^{2} \)
\( J_2 \)  = \(-60\) =  \( - 2^{2} \cdot 3 \cdot 5 \)
\( J_4 \)  = \(104\) =  \( 2^{3} \cdot 13 \)
\( J_6 \)  = \(-504\) =  \( - 2^{3} \cdot 3^{2} \cdot 7 \)
\( J_8 \)  = \(4856\) =  \( 2^{3} \cdot 607 \)
\( J_{10} \)  = \(-847\) =  \( - 7 \cdot 11^{2} \)
\( g_1 \)  = \(777600000/847\)
\( g_2 \)  = \(22464000/847\)
\( g_3 \)  = \(259200/121\)

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

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

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],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: \(\Z \times \Z/{5}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.196055\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(5\)

2-torsion field: 6.0.54208.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.196055 \)
Real period: \( 20.30596 \)
Tamagawa product: \( 1 \)
Torsion order:\( 5 \)
Leading coefficient: \( 0.159244 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш: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.a1
  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 \(\Q\).