Properties

Label 72900.b.291600.1
Conductor $72900$
Discriminant $-291600$
Mordell-Weil group \(\Z/{3}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = 2x^3 + 5$ (homogenize, simplify)
$y^2 + x^3y = 2x^3z^3 + 5z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^3 + 20$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 0, 0, 2]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, 0, 0, 2], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([20, 0, 0, 8, 0, 0, 1]))
 
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: \((1 : 0 : 0),\, (1 : -1 : 0)\)

magma: [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: \(\Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.0.4665600.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 10.31518 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 1.146131 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 \(\Q (b) \simeq \) 6.2.36450000.1 with defining polynomial:
  \(x^{6} - 20\)

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
  \(g_4 = \frac{41648544}{130321} b^{5} + \frac{69524280}{130321} b^{4} + \frac{116946000}{130321} b^{3} + \frac{191074500}{130321} b^{2} + \frac{317836800}{130321} b + \frac{523159920}{130321}\)
  \(g_6 = -\frac{2379059877600}{47045881} b^{5} - \frac{3914934599520}{47045881} b^{4} - \frac{6448175385120}{47045881} b^{3} - \frac{10640543492880}{47045881} b^{2} - \frac{17508481814880}{47045881} b - \frac{28834939662480}{47045881}\)
   Conductor norm: 729
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{41648544}{130321} b^{5} + \frac{69524280}{130321} b^{4} - \frac{116946000}{130321} b^{3} + \frac{191074500}{130321} b^{2} - \frac{317836800}{130321} b + \frac{523159920}{130321}\)
  \(g_6 = \frac{2379059877600}{47045881} b^{5} - \frac{3914934599520}{47045881} b^{4} + \frac{6448175385120}{47045881} b^{3} - \frac{10640543492880}{47045881} b^{2} + \frac{17508481814880}{47045881} b - \frac{28834939662480}{47045881}\)
   Conductor norm: 729

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 \(x^{12} - 6 x^{11} + 21 x^{10} - 50 x^{9} + 90 x^{8} - 126 x^{7} + 101 x^{6} - 6 x^{5} + 390 x^{4} - 850 x^{3} + 321 x^{2} + 114 x + 361\)

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 \(\frac{7868}{3501453} a^{11} - \frac{43274}{3501453} a^{10} + \frac{152740}{3501453} a^{9} - \frac{120925}{1167151} a^{8} + \frac{660700}{3501453} a^{7} - \frac{922418}{3501453} a^{6} + \frac{671972}{3501453} a^{5} - \frac{70}{61429} a^{4} + \frac{884940}{1167151} a^{3} - \frac{4262510}{3501453} a^{2} + \frac{2701228}{3501453} a + \frac{59134}{184287}\) 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 \(F \simeq \) \(\Q(\sqrt{-15}) \) with generator \(\frac{1}{228} a^{10} - \frac{5}{228} a^{9} + \frac{4}{57} a^{8} - \frac{17}{114} a^{7} + \frac{14}{57} a^{6} - \frac{35}{114} a^{5} + \frac{25}{114} a^{4} - \frac{13}{228} a^{3} + \frac{491}{228} a^{2} - \frac{41}{19} a + \frac{1}{4}\) 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 \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(-\frac{189}{122858} a^{11} + \frac{2079}{245716} a^{10} - \frac{6245}{245716} a^{9} + \frac{6255}{122858} a^{8} - \frac{8001}{122858} a^{7} + \frac{3045}{61429} a^{6} + \frac{11403}{122858} a^{5} - \frac{34335}{122858} a^{4} - \frac{119019}{245716} a^{3} + \frac{248067}{245716} a^{2} + \frac{92511}{122858} a - \frac{13317}{245716}\) 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 \(F \simeq \) 3.1.2700.1 with generator \(\frac{14}{2679} a^{10} - \frac{70}{2679} a^{9} + \frac{205}{2679} a^{8} - \frac{400}{2679} a^{7} + \frac{518}{2679} a^{6} - \frac{448}{2679} a^{5} - \frac{210}{893} a^{4} + \frac{560}{893} a^{3} + \frac{6209}{2679} a^{2} - \frac{7078}{2679} a - \frac{274}{141}\) with minimal polynomial \(x^{3} - 20\):

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

Over subfield \(F \simeq \) 3.1.2700.1 with generator \(-\frac{3248}{3501453} a^{11} + \frac{2905}{1167151} a^{10} + \frac{3460}{3501453} a^{9} - \frac{77665}{3501453} a^{8} + \frac{95904}{1167151} a^{7} - \frac{189434}{1167151} a^{6} + \frac{1062068}{3501453} a^{5} - \frac{298340}{1167151} a^{4} - \frac{864680}{3501453} a^{3} - \frac{1075430}{1167151} a^{2} + \frac{4330986}{1167151} a - \frac{50407}{184287}\) with minimal polynomial \(x^{3} - 20\):

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

Over subfield \(F \simeq \) 3.1.2700.1 with generator \(\frac{3248}{3501453} a^{11} - \frac{27013}{3501453} a^{10} + \frac{88030}{3501453} a^{9} - \frac{190270}{3501453} a^{8} + \frac{235088}{3501453} a^{7} - \frac{108724}{3501453} a^{6} - \frac{158844}{1167151} a^{5} + \frac{572810}{1167151} a^{4} - \frac{1331080}{3501453} a^{3} - \frac{4888873}{3501453} a^{2} - \frac{196948}{184287} a + \frac{136175}{61429}\) with minimal polynomial \(x^{3} - 20\):

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

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}, \sqrt{5})\) with generator \(\frac{4963}{14005812} a^{11} + \frac{1709}{7002906} a^{10} - \frac{26075}{14005812} a^{9} + \frac{6443}{737148} a^{8} - \frac{29825}{2334302} a^{7} + \frac{37051}{2334302} a^{6} - \frac{39025}{3501453} a^{5} - \frac{214675}{7002906} a^{4} + \frac{759155}{7002906} a^{3} + \frac{4541903}{4668604} a^{2} - \frac{4435951}{14005812} a - \frac{89069}{368574}\) 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 \(F \simeq \) 6.0.21870000.2 with generator \(-\frac{14197}{10504359} a^{11} + \frac{54221}{21008718} a^{10} + \frac{19393}{3501453} a^{9} - \frac{127107}{2334302} a^{8} + \frac{1867481}{10504359} a^{7} - \frac{8351567}{21008718} a^{6} + \frac{7197245}{10504359} a^{5} - \frac{14678867}{21008718} a^{4} - \frac{3522077}{10504359} a^{3} - \frac{18678211}{21008718} a^{2} + \frac{4717505}{1167151} a - \frac{46481}{23526}\) with minimal polynomial \(x^{6} - 9 x^{4} - 20 x^{3} + 147 x^{2} - 240 x + 133\):

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

Over subfield \(F \simeq \) 6.2.36450000.1 with generator \(\frac{3934}{3501453} a^{11} - \frac{12488}{3501453} a^{10} + \frac{30625}{3501453} a^{9} - \frac{47420}{3501453} a^{8} + \frac{68950}{3501453} a^{7} - \frac{122696}{3501453} a^{6} + \frac{14406}{1167151} a^{5} - \frac{137900}{1167151} a^{4} + \frac{808430}{1167151} a^{3} + \frac{175600}{3501453} a^{2} + \frac{11926}{184287} a - \frac{333779}{184287}\) with minimal polynomial \(x^{6} - 20\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.0.109350000.2 with generator \(\frac{35239}{7002906} a^{11} - \frac{21227}{737148} a^{10} + \frac{342958}{3501453} a^{9} - \frac{3226163}{14005812} a^{8} + \frac{149735}{368574} a^{7} - \frac{8045429}{14005812} a^{6} + \frac{3243941}{7002906} a^{5} - \frac{540909}{4668604} a^{4} + \frac{5186585}{2334302} a^{3} - \frac{18055781}{4668604} a^{2} + \frac{5899774}{3501453} a - \frac{826145}{737148}\) with minimal polynomial \(x^{6} - 10 x^{3} + 160\):

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

Over subfield \(F \simeq \) 6.0.109350000.2 with generator \(-\frac{11635}{7002906} a^{11} + \frac{25235}{4668604} a^{10} - \frac{9631}{1167151} a^{9} - \frac{18735}{2334302} a^{8} + \frac{441521}{7002906} a^{7} - \frac{2338049}{14005812} a^{6} + \frac{2193701}{7002906} a^{5} - \frac{1756445}{7002906} a^{4} - \frac{7004531}{7002906} a^{3} + \frac{3118853}{14005812} a^{2} + \frac{19653691}{7002906} a - \frac{223133}{368574}\) with minimal polynomial \(x^{6} - 10 x^{3} + 160\):

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

Over subfield \(F \simeq \) 6.0.109350000.2 with generator \(\frac{2}{893} a^{10} - \frac{10}{893} a^{9} + \frac{479}{10716} a^{8} - \frac{299}{2679} a^{7} + \frac{1337}{5358} a^{6} - \frac{1085}{2679} a^{5} + \frac{5171}{10716} a^{4} - \frac{1066}{2679} a^{3} + \frac{6215}{5358} a^{2} - \frac{5429}{5358} a + \frac{1229}{564}\) with minimal polynomial \(x^{6} - 10 x^{3} + 160\):

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

Over subfield \(F \simeq \) 6.2.36450000.1 with generator \(\frac{7868}{3501453} a^{11} - \frac{43274}{3501453} a^{10} + \frac{152740}{3501453} a^{9} - \frac{120925}{1167151} a^{8} + \frac{660700}{3501453} a^{7} - \frac{922418}{3501453} a^{6} + \frac{671972}{3501453} a^{5} - \frac{70}{61429} a^{4} + \frac{884940}{1167151} a^{3} - \frac{4262510}{3501453} a^{2} + \frac{6202681}{3501453} a - \frac{125153}{184287}\) with minimal polynomial \(x^{6} - 20\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.2.36450000.1 with generator \(-\frac{3934}{3501453} a^{11} + \frac{10262}{1167151} a^{10} - \frac{40705}{1167151} a^{9} + \frac{315355}{3501453} a^{8} - \frac{197250}{1167151} a^{7} + \frac{266574}{1167151} a^{6} - \frac{628754}{3501453} a^{5} - \frac{136570}{1167151} a^{4} - \frac{76510}{1167151} a^{3} + \frac{1479370}{1167151} a^{2} - \frac{1992029}{1167151} a - \frac{69542}{61429}\) with minimal polynomial \(x^{6} - 20\):

\(\End (J_{F})\)\(\simeq\)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