Properties

Label 18225.a.18225.1
Conductor $18225$
Discriminant $-18225$
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

Related objects

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Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(18225\) \(=\) \( 3^{6} \cdot 5^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-18225\) \(=\) \( - 3^{6} \cdot 5^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(196\) \(=\)  \( 2^{2} \cdot 7^{2} \)
\( I_4 \)  \(=\) \(1305\) \(=\)  \( 3^{2} \cdot 5 \cdot 29 \)
\( I_6 \)  \(=\) \(73965\) \(=\)  \( 3 \cdot 5 \cdot 4931 \)
\( I_{10} \)  \(=\) \(9600\) \(=\)  \( 2^{7} \cdot 3 \cdot 5^{2} \)
\( J_2 \)  \(=\) \(147\) \(=\)  \( 3 \cdot 7^{2} \)
\( J_4 \)  \(=\) \(411\) \(=\)  \( 3 \cdot 137 \)
\( J_6 \)  \(=\) \(-401\) \(=\)  \( -401 \)
\( J_8 \)  \(=\) \(-56967\) \(=\)  \( - 3 \cdot 17 \cdot 1117 \)
\( J_{10} \)  \(=\) \(18225\) \(=\)  \( 3^{6} \cdot 5^{2} \)
\( g_1 \)  \(=\) \(282475249/75\)
\( g_2 \)  \(=\) \(16117913/225\)
\( g_3 \)  \(=\) \(-962801/2025\)

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

magma: [C![-1,-1,1],C![-1,1,1],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: \(\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.687920\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(3\)

2-torsion field: 6.0.1166400.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.687920 \)
Real period: \( 16.46817 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 1.258756 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 \(\Q (b) \simeq \) 6.2.2278125.1 with defining polynomial:
  \(x^{6} - 5\)

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
  \(g_4 = 36 b^{5} + 225 b^{2}\)
  \(g_6 = -1890 b^{3} - 7695\)
   Conductor norm: 729
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -36 b^{5} + 225 b^{2}\)
  \(g_6 = 1890 b^{3} - 7695\)
   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 \(x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 195 x^{8} - 366 x^{7} + 551 x^{6} - 642 x^{5} + 585 x^{4} - 400 x^{3} + 177 x^{2} - 42 x + 4\)

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 \(\frac{433}{184} a^{11} - \frac{4763}{368} a^{10} + \frac{21035}{368} a^{9} - \frac{58935}{368} a^{8} + \frac{70075}{184} a^{7} - \frac{62209}{92} a^{6} + \frac{178651}{184} a^{5} - \frac{384865}{368} a^{4} + \frac{324585}{368} a^{3} - \frac{194285}{368} a^{2} + \frac{32181}{184} a - \frac{2031}{92}\) 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 \(\frac{93}{92} a^{11} - \frac{1023}{184} a^{10} + \frac{4499}{184} a^{9} - \frac{12573}{184} a^{8} + \frac{14871}{92} a^{7} - \frac{6573}{23} a^{6} + \frac{37407}{92} a^{5} - \frac{79797}{184} a^{4} + \frac{66117}{184} a^{3} - \frac{38823}{184} a^{2} + \frac{5841}{92} a - \frac{257}{46}\) 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 \(-\frac{19}{16} a^{10} + \frac{95}{16} a^{9} - \frac{413}{16} a^{8} + \frac{541}{8} a^{7} - \frac{629}{4} a^{6} + 260 a^{5} - \frac{5673}{16} a^{4} + \frac{5485}{16} a^{3} - \frac{4255}{16} a^{2} + \frac{1027}{8} a - \frac{85}{4}\) 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 \(F \simeq \) 3.1.675.1 with generator \(\frac{291}{92} a^{11} - \frac{6241}{368} a^{10} + \frac{27445}{368} a^{9} - \frac{75591}{368} a^{8} + \frac{89331}{184} a^{7} - \frac{77923}{92} a^{6} + \frac{110397}{92} a^{5} - \frac{463635}{368} a^{4} + \frac{381951}{368} a^{3} - \frac{218125}{368} a^{2} + \frac{31917}{184} a - \frac{1615}{92}\) with minimal polynomial \(x^{3} - 5\):

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

Over subfield \(F \simeq \) 3.1.675.1 with generator \(-\frac{291}{92} a^{11} + \frac{6563}{368} a^{10} - \frac{29055}{368} a^{9} + \frac{82629}{368} a^{8} - \frac{98577}{184} a^{7} + \frac{88733}{92} a^{6} - \frac{128337}{92} a^{5} + \frac{562305}{368} a^{4} - \frac{478045}{368} a^{3} + \frac{292967}{368} a^{2} - \frac{49995}{184} a + \frac{3225}{92}\) with minimal polynomial \(x^{3} - 5\):

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

Over subfield \(F \simeq \) 3.1.675.1 with generator \(-\frac{7}{8} a^{10} + \frac{35}{8} a^{9} - \frac{153}{8} a^{8} + \frac{201}{4} a^{7} - \frac{235}{2} a^{6} + 195 a^{5} - \frac{2145}{8} a^{4} + \frac{2089}{8} a^{3} - \frac{1627}{8} a^{2} + \frac{393}{4} a - \frac{35}{2}\) with minimal polynomial \(x^{3} - 5\):

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

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}, \sqrt{5})\) with generator \(\frac{247}{368} a^{11} - \frac{285}{92} a^{10} + \frac{2463}{184} a^{9} - \frac{12145}{368} a^{8} + \frac{13945}{184} a^{7} - \frac{10725}{92} a^{6} + \frac{55997}{368} a^{5} - \frac{11849}{92} a^{4} + \frac{16549}{184} a^{3} - \frac{9387}{368} a^{2} - \frac{1561}{184} a + \frac{265}{92}\) 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 \(F \simeq \) 6.0.1366875.1 with generator \(-\frac{149}{184} a^{11} + \frac{739}{184} a^{10} - \frac{3205}{184} a^{9} + \frac{1041}{23} a^{8} - \frac{2407}{23} a^{7} + \frac{7857}{46} a^{6} - \frac{42143}{184} a^{5} + \frac{39385}{184} a^{4} - \frac{28683}{184} a^{3} + \frac{1490}{23} a^{2} + \frac{33}{23} a - \frac{104}{23}\) with minimal polynomial \(x^{6} - 3 x^{5} + 6 x^{4} + 3 x^{3} - 9 x^{2} - 18 x + 36\):

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

Over subfield \(F \simeq \) 6.0.6834375.1 with generator \(\frac{591}{368} a^{11} - \frac{201}{23} a^{10} + \frac{7071}{184} a^{9} - \frac{39289}{368} a^{8} + \frac{46449}{184} a^{7} - \frac{40837}{92} a^{6} + \frac{231869}{368} a^{5} - \frac{15346}{23} a^{4} + \frac{101237}{184} a^{3} - \frac{116763}{368} a^{2} + \frac{17035}{184} a - \frac{699}{92}\) with minimal polynomial \(x^{6} - 3 x^{5} - 5 x^{3} + 15 x^{2} + 12 x + 4\):

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

Over subfield \(F \simeq \) 6.2.2278125.1 with generator \(\frac{167}{92} a^{11} - \frac{3789}{368} a^{10} + \frac{16797}{368} a^{9} - \frac{47951}{368} a^{8} + \frac{57311}{184} a^{7} - \frac{51791}{92} a^{6} + \frac{75195}{92} a^{5} - \frac{331527}{368} a^{4} + \frac{284023}{368} a^{3} - \frac{176605}{368} a^{2} + \frac{31153}{184} a - \frac{2187}{92}\) with minimal polynomial \(x^{6} - 5\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.2.2278125.1 with generator \(\frac{167}{46} a^{11} - \frac{1837}{92} a^{10} + \frac{8111}{92} a^{9} - \frac{11361}{46} a^{8} + \frac{27011}{46} a^{7} - \frac{23975}{23} a^{6} + \frac{34412}{23} a^{5} - \frac{148203}{92} a^{4} + \frac{124865}{92} a^{3} - \frac{18663}{23} a^{2} + \frac{12207}{46} a - \frac{760}{23}\) with minimal polynomial \(x^{6} - 5\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.2.2278125.1 with generator \(\frac{167}{92} a^{11} - \frac{3559}{368} a^{10} + \frac{15647}{368} a^{9} - \frac{42937}{368} a^{8} + \frac{50733}{184} a^{7} - \frac{44109}{92} a^{6} + \frac{62453}{92} a^{5} - \frac{261285}{368} a^{4} + \frac{215437}{368} a^{3} - \frac{122003}{368} a^{2} + \frac{17675}{184} a - \frac{853}{92}\) with minimal polynomial \(x^{6} - 5\):

\(\End (J_{F})\)\(\simeq\)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 \(F \simeq \) 6.0.6834375.1 with generator \(-\frac{573}{368} a^{11} + \frac{739}{92} a^{10} - \frac{6479}{184} a^{9} + \frac{34807}{368} a^{8} - \frac{40927}{184} a^{7} + \frac{34809}{92} a^{6} - \frac{194631}{368} a^{5} + \frac{49321}{92} a^{4} - \frac{79561}{184} a^{3} + \frac{84549}{368} a^{2} - \frac{10581}{184} a + \frac{433}{92}\) with minimal polynomial \(x^{6} - 3 x^{5} - 5 x^{3} + 15 x^{2} + 12 x + 4\):

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

Over subfield \(F \simeq \) 6.0.6834375.1 with generator \(-\frac{1}{8} a^{10} + \frac{5}{8} a^{9} - \frac{21}{8} a^{8} + \frac{27}{4} a^{7} - 15 a^{6} + 24 a^{5} - \frac{239}{8} a^{4} + \frac{211}{8} a^{3} - \frac{135}{8} a^{2} + \frac{27}{4} a + \frac{3}{2}\) with minimal polynomial \(x^{6} - 3 x^{5} - 5 x^{3} + 15 x^{2} + 12 x + 4\):

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