Properties

Label 43264.c.43264.1
Conductor $43264$
Discriminant $43264$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $E_6$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(43264\) \(=\) \( 2^{8} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(43264\) \(=\) \( 2^{8} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(110\) \(=\)  \( 2 \cdot 5 \cdot 11 \)
\( I_4 \)  \(=\) \(520\) \(=\)  \( 2^{3} \cdot 5 \cdot 13 \)
\( I_6 \)  \(=\) \(15470\) \(=\)  \( 2 \cdot 5 \cdot 7 \cdot 13 \cdot 17 \)
\( I_{10} \)  \(=\) \(169\) \(=\)  \( 13^{2} \)
\( J_2 \)  \(=\) \(220\) \(=\)  \( 2^{2} \cdot 5 \cdot 11 \)
\( J_4 \)  \(=\) \(630\) \(=\)  \( 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
\( J_6 \)  \(=\) \(-620\) \(=\)  \( - 2^{2} \cdot 5 \cdot 31 \)
\( J_8 \)  \(=\) \(-133325\) \(=\)  \( - 5^{2} \cdot 5333 \)
\( J_{10} \)  \(=\) \(43264\) \(=\)  \( 2^{8} \cdot 13^{2} \)
\( g_1 \)  \(=\) \(2013137500/169\)
\( g_2 \)  \(=\) \(52408125/338\)
\( g_3 \)  \(=\) \(-468875/676\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_6$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

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

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 3.3.169.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 23.10504 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.444065 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(8\) \(1\) \(1\)
\(13\) \(2\) \(2\) \(1\) \(1 + 2 T + 13 T^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_6$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial:
  \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 196880 b^{5} - 67520 b^{4} - 1012640 b^{3} + 97360 b^{2} + 1187280 b + 267120\)
  \(g_6 = 117255840 b^{5} - 32032000 b^{4} - 613207296 b^{3} + 28686944 b^{2} + 737461088 b + 167732032\)
   Conductor norm: 16777216

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\) with minimal polynomial \(x^{2} - x - 3\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.169.1 with generator \(a^{3} - a^{2} - 3 a + 2\) with minimal polynomial \(x^{3} - x^{2} - 4 x - 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple