Properties

Label 72900.a.291600.1
Conductor $72900$
Discriminant $-291600$
Mordell-Weil group \(\Z \times \Z/{3}\Z\)
Sato-Tate group $J(E_6)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 0, 0, -2, 0, 0, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 0, 0, -2, 0, 0, 1], R![1]);
 
sage: X = HyperellipticCurve(R([5, 0, 0, -8, 0, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(72900\) \(=\) \( 2^{2} \cdot 3^{6} \cdot 5^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-291600\) \(=\) \( - 2^{4} \cdot 3^{6} \cdot 5^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(184\) \(=\)  \( 2^{3} \cdot 23 \)
\( I_4 \)  \(=\) \(1845\) \(=\)  \( 3^{2} \cdot 5 \cdot 41 \)
\( I_6 \)  \(=\) \(87015\) \(=\)  \( 3 \cdot 5 \cdot 5801 \)
\( I_{10} \)  \(=\) \(150\) \(=\)  \( 2 \cdot 3 \cdot 5^{2} \)
\( J_2 \)  \(=\) \(552\) \(=\)  \( 2^{3} \cdot 3 \cdot 23 \)
\( J_4 \)  \(=\) \(1626\) \(=\)  \( 2 \cdot 3 \cdot 271 \)
\( J_6 \)  \(=\) \(-1616\) \(=\)  \( - 2^{4} \cdot 101 \)
\( J_8 \)  \(=\) \(-883977\) \(=\)  \( - 3 \cdot 294659 \)
\( J_{10} \)  \(=\) \(291600\) \(=\)  \( 2^{4} \cdot 3^{6} \cdot 5^{2} \)
\( g_1 \)  \(=\) \(13181630464/75\)
\( g_2 \)  \(=\) \(211024448/225\)
\( g_3 \)  \(=\) \(-3419456/2025\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
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: \((1 : -1 : 0),\, (1 : 1 : 0),\, (1 : 0 : 1),\, (1 : -1 : 1)\)

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

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2z^3\) \(0.563877\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0\) \(3\)

2-torsion field: 6.0.4665600.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.563877 \)
Real period: \( 10.31518 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 1.938832 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(3\) \(1 + 2 T^{2}\)
\(3\) \(6\) \(6\) \(1\) \(1\)
\(5\) \(2\) \(2\) \(1\) \(1 + 5 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.2.450000.1 with defining polynomial:
  \(x^{6} - x^{5} - 5 x^{3} - x + 1\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 192 b^{5} - 690 b^{4} + 690 b^{3} - 750 b^{2} + 2280 b - 672\)
  \(g_6 = -20160 b^{5} + 10080 b^{4} + 20160 b^{3} + 80640 b^{2} + 40320 b - 39060\)
   Conductor norm: 4782969
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -192 b^{5} - 210 b^{4} + 210 b^{3} + 1650 b^{2} + 1320 b + 672\)
  \(g_6 = 20160 b^{5} - 10080 b^{4} - 20160 b^{3} - 80640 b^{2} - 40320 b - 28980\)
   Conductor norm: 4782969

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a)\) with defining polynomial \(x^{12} - x^{11} + x^{10} + 10 x^{9} - 5 x^{8} - x^{7} + 26 x^{6} - x^{5} - 5 x^{4} + 10 x^{3} + x^{2} - x + 1\)

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 \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(\frac{13}{12} a^{11} - \frac{7}{6} a^{10} + \frac{5}{4} a^{9} + \frac{125}{12} a^{8} - \frac{35}{6} a^{7} - \frac{1}{4} a^{6} + \frac{307}{12} a^{5} - \frac{5}{3} a^{4} - \frac{15}{4} a^{3} + \frac{65}{12} a^{2} + \frac{2}{3} a + \frac{1}{4}\) with minimal polynomial \(x^{2} - x + 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_6$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(\frac{1}{36} a^{11} - \frac{1}{2} a^{10} + \frac{7}{36} a^{9} + \frac{5}{36} a^{8} - \frac{31}{6} a^{7} - \frac{19}{36} a^{6} + \frac{103}{36} a^{5} - \frac{35}{3} a^{4} - \frac{221}{36} a^{3} + \frac{101}{36} a^{2} - \frac{7}{3} a - \frac{25}{36}\) with minimal polynomial \(x^{2} - x - 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_3)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-15}) \) with generator \(-\frac{1}{3} a^{11} + a^{10} - \frac{4}{3} a^{9} - 2 a^{8} + \frac{23}{3} a^{7} - 6 a^{6} - \frac{13}{3} a^{5} + 16 a^{4} - \frac{22}{3} a^{3} + a^{2} + \frac{17}{3} a - 1\) with minimal polynomial \(x^{2} - x + 4\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_3)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.300.1 with generator \(-\frac{1}{6} a^{11} + \frac{7}{6} a^{10} - a^{9} - \frac{5}{6} a^{8} + \frac{65}{6} a^{7} - 3 a^{6} - \frac{37}{6} a^{5} + \frac{145}{6} a^{4} + 5 a^{3} - \frac{35}{6} a^{2} + \frac{29}{6} a + 2\) with minimal polynomial \(x^{3} - x^{2} - 3 x - 3\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.300.1 with generator \(\frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{2} a^{9} + \frac{31}{6} a^{8} - \frac{17}{6} a^{7} - \frac{1}{6} a^{6} + \frac{85}{6} a^{5} - \frac{5}{2} a^{4} - \frac{13}{6} a^{3} + \frac{43}{6} a^{2} - \frac{7}{6} a - \frac{1}{6}\) with minimal polynomial \(x^{3} - x^{2} - 3 x - 3\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.300.1 with generator \(-\frac{1}{3} a^{11} - \frac{2}{3} a^{10} + \frac{1}{2} a^{9} - \frac{13}{3} a^{8} - 8 a^{7} + \frac{19}{6} a^{6} - 8 a^{5} - \frac{65}{3} a^{4} - \frac{17}{6} a^{3} - \frac{4}{3} a^{2} - \frac{11}{3} a - \frac{5}{6}\) with minimal polynomial \(x^{3} - x^{2} - 3 x - 3\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}, \sqrt{5})\) with generator \(-\frac{7}{18} a^{11} + \frac{1}{3} a^{10} - \frac{1}{18} a^{9} - \frac{77}{18} a^{8} + \frac{5}{3} a^{7} + \frac{61}{18} a^{6} - \frac{217}{18} a^{5} - \frac{4}{3} a^{4} + \frac{155}{18} a^{3} - \frac{83}{18} a^{2} - 2 a + \frac{31}{18}\) with minimal polynomial \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

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

Over subfield \(F \simeq \) 6.0.270000.1 with generator \(\frac{5}{12} a^{11} - \frac{1}{3} a^{10} + \frac{1}{12} a^{9} + \frac{53}{12} a^{8} - \frac{4}{3} a^{7} - \frac{41}{12} a^{6} + \frac{45}{4} a^{5} + \frac{19}{6} a^{4} - \frac{33}{4} a^{3} + \frac{29}{12} a^{2} + \frac{7}{2} a - \frac{11}{12}\) with minimal polynomial \(x^{6} - 3 x^{5} + 4 x^{4} - 3 x^{3} - 2 x^{2} + 3 x + 3\):

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

Over subfield \(F \simeq \) 6.0.1350000.1 with generator \(\frac{1}{9} a^{11} - \frac{5}{6} a^{10} + \frac{10}{9} a^{9} - \frac{1}{9} a^{8} - \frac{43}{6} a^{7} + \frac{53}{9} a^{6} + \frac{1}{9} a^{5} - \frac{101}{6} a^{4} + \frac{55}{9} a^{3} - \frac{10}{9} a^{2} - \frac{31}{6} a + \frac{8}{9}\) with minimal polynomial \(x^{6} - x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 6 x + 6\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 6.2.450000.1 with generator \(-\frac{23}{36} a^{11} + \frac{1}{2} a^{10} - \frac{11}{36} a^{9} - \frac{235}{36} a^{8} + \frac{11}{6} a^{7} + \frac{119}{36} a^{6} - \frac{557}{36} a^{5} - 4 a^{4} + \frac{253}{36} a^{3} - \frac{55}{36} a^{2} - 2 a + \frac{29}{36}\) with minimal polynomial \(x^{6} - x^{5} - 5 x^{3} - x + 1\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.2.450000.1 with generator \(\frac{31}{36} a^{11} - \frac{4}{3} a^{10} + \frac{55}{36} a^{9} + \frac{275}{36} a^{8} - \frac{25}{3} a^{7} + \frac{101}{36} a^{6} + \frac{697}{36} a^{5} - \frac{61}{6} a^{4} - \frac{17}{36} a^{3} + \frac{179}{36} a^{2} - \frac{5}{6} a - \frac{1}{36}\) with minimal polynomial \(x^{6} - x^{5} - 5 x^{3} - x + 1\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.2.450000.1 with generator \(\frac{1}{12} a^{11} - \frac{1}{6} a^{10} + \frac{5}{12} a^{9} + \frac{5}{12} a^{8} - \frac{5}{6} a^{7} + \frac{31}{12} a^{6} + \frac{7}{12} a^{5} - \frac{5}{3} a^{4} + \frac{65}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3} a + \frac{13}{12}\) with minimal polynomial \(x^{6} - x^{5} - 5 x^{3} - x + 1\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 6.0.1350000.1 with generator \(-\frac{1}{2} a^{11} + \frac{5}{6} a^{10} - a^{9} - \frac{9}{2} a^{8} + \frac{11}{2} a^{7} - \frac{8}{3} a^{6} - \frac{25}{2} a^{5} + \frac{15}{2} a^{4} - a^{3} - \frac{31}{6} a^{2} - \frac{1}{2} a\) with minimal polynomial \(x^{6} - x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 6 x + 6\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 6.0.1350000.1 with generator \(-\frac{1}{18} a^{11} + \frac{1}{2} a^{10} - \frac{13}{18} a^{9} + \frac{7}{18} a^{8} + \frac{25}{6} a^{7} - \frac{71}{18} a^{6} + \frac{29}{18} a^{5} + \frac{61}{6} a^{4} - \frac{97}{18} a^{3} + \frac{49}{18} a^{2} + \frac{23}{6} a - \frac{23}{18}\) with minimal polynomial \(x^{6} - x^{5} + 5 x^{4} - 5 x^{3} + 10 x^{2} - 6 x + 6\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple