Properties

Label 644.a.2576.1
Conductor $644$
Discriminant $-2576$
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = -5x^6 + 11x^5 - 20x^4 + 20x^3 - 20x^2 + 11x - 5$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = -5x^6 + 11x^5z - 20x^4z^2 + 20x^3z^3 - 20x^2z^4 + 11xz^5 - 5z^6$ (dehomogenize, simplify)
$y^2 = -20x^6 + 44x^5 - 79x^4 + 82x^3 - 79x^2 + 44x - 20$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 11, -20, 20, -20, 11, -5]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 11, -20, 20, -20, 11, -5], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-20, 44, -79, 82, -79, 44, -20]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(644\) \(=\) \( 2^{2} \cdot 7 \cdot 23 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-2576\) \(=\) \( - 2^{4} \cdot 7 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(39036\) \(=\)  \( 2^{2} \cdot 3 \cdot 3253 \)
\( I_4 \)  \(=\) \(4124865\) \(=\)  \( 3 \cdot 5 \cdot 73 \cdot 3767 \)
\( I_6 \)  \(=\) \(50880984159\) \(=\)  \( 3 \cdot 73 \cdot 643 \cdot 361327 \)
\( I_{10} \)  \(=\) \(329728\) \(=\)  \( 2^{11} \cdot 7 \cdot 23 \)
\( J_2 \)  \(=\) \(9759\) \(=\)  \( 3 \cdot 3253 \)
\( J_4 \)  \(=\) \(3796384\) \(=\)  \( 2^{5} \cdot 31 \cdot 43 \cdot 89 \)
\( J_6 \)  \(=\) \(1910683600\) \(=\)  \( 2^{4} \cdot 5^{2} \cdot 7 \cdot 23 \cdot 29669 \)
\( J_8 \)  \(=\) \(1058457444236\) \(=\)  \( 2^{2} \cdot 269891 \cdot 980449 \)
\( J_{10} \)  \(=\) \(2576\) \(=\)  \( 2^{4} \cdot 7 \cdot 23 \)
\( g_1 \)  \(=\) \(88516980336138032799/2576\)
\( g_2 \)  \(=\) \(220529201888022246/161\)
\( g_3 \)  \(=\) \(70640465629725\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
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: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(6\)

2-torsion field: 8.0.1698758656.7

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 3.928431 \)
Tamagawa product: \( 1 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.218246 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(1\) \(( 1 + T )^{2}\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 23 T^{2} )\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 14.a4
  Elliptic curve 46.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 \(\Q\).