Global number field 36.0.48526755753740305052512669329205843844959387036328330042655969.1

• Label: 36.0.48526755753...5969.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $3^{90}\cdot 11^{18}$
• Root discriminant: $51.70$
• Ramified primes: $3, 11$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - 136 x^{27} - 1187 x^{18} - 2676888 x^{9} + 387420489 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(48526755753740305052512669329205843844959387036328330042655969=3^{90}\cdot 11^{18}\)
Root discriminant:		$51.70$
Ramified primes:		$3, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(297=3^{3}\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{297}(1,·)$, $\chi_{297}(131,·)$, $\chi_{297}(133,·)$, $\chi_{297}(263,·)$, $\chi_{297}(265,·)$, $\chi_{297}(10,·)$, $\chi_{297}(142,·)$, $\chi_{297}(274,·)$, $\chi_{297}(23,·)$, $\chi_{297}(155,·)$, $\chi_{297}(287,·)$, $\chi_{297}(32,·)$, $\chi_{297}(34,·)$, $\chi_{297}(164,·)$, $\chi_{297}(166,·)$, $\chi_{297}(296,·)$, $\chi_{297}(43,·)$, $\chi_{297}(175,·)$, $\chi_{297}(56,·)$, $\chi_{297}(188,·)$, $\chi_{297}(65,·)$, $\chi_{297}(67,·)$, $\chi_{297}(197,·)$, $\chi_{297}(199,·)$, $\chi_{297}(76,·)$, $\chi_{297}(208,·)$, $\chi_{297}(89,·)$, $\chi_{297}(221,·)$, $\chi_{297}(98,·)$, $\chi_{297}(100,·)$, $\chi_{297}(230,·)$, $\chi_{297}(232,·)$, $\chi_{297}(109,·)$, $\chi_{297}(241,·)$, $\chi_{297}(122,·)$, $\chi_{297}(254,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{74} a^{18} - \frac{31}{74} a^{9} - \frac{1}{74}$, $\frac{1}{222} a^{19} - \frac{31}{222} a^{10} + \frac{73}{222} a$, $\frac{1}{666} a^{20} - \frac{253}{666} a^{11} + \frac{73}{666} a^{2}$, $\frac{1}{1998} a^{21} - \frac{919}{1998} a^{12} - \frac{593}{1998} a^{3}$, $\frac{1}{5994} a^{22} + \frac{1079}{5994} a^{13} + \frac{1405}{5994} a^{4}$, $\frac{1}{17982} a^{23} + \frac{1079}{17982} a^{14} - \frac{4589}{17982} a^{5}$, $\frac{1}{53946} a^{24} - \frac{16903}{53946} a^{15} + \frac{13393}{53946} a^{6}$, $\frac{1}{161838} a^{25} + \frac{37043}{161838} a^{16} - \frac{40553}{161838} a^{7}$, $\frac{1}{485514} a^{26} - \frac{124795}{485514} a^{17} - \frac{40553}{485514} a^{8}$, $\frac{1}{1728915354} a^{27} - \frac{796}{728271} a^{18} + \frac{49207}{728271} a^{9} - \frac{8445}{87838}$, $\frac{1}{5186746062} a^{28} - \frac{796}{2184813} a^{19} - \frac{679064}{2184813} a^{10} - \frac{96283}{263514} a$, $\frac{1}{15560238186} a^{29} - \frac{796}{6554439} a^{20} - \frac{679064}{6554439} a^{11} + \frac{167231}{790542} a^{2}$, $\frac{1}{46680714558} a^{30} - \frac{796}{19663317} a^{21} - \frac{679064}{19663317} a^{12} - \frac{623311}{2371626} a^{3}$, $\frac{1}{140042143674} a^{31} - \frac{796}{58989951} a^{22} + \frac{18984253}{58989951} a^{13} + \frac{1748315}{7114878} a^{4}$, $\frac{1}{420126431022} a^{32} - \frac{796}{176969853} a^{23} + \frac{18984253}{176969853} a^{14} - \frac{5366563}{21344634} a^{5}$, $\frac{1}{1260379293066} a^{33} - \frac{796}{530909559} a^{24} + \frac{18984253}{530909559} a^{15} - \frac{26711197}{64033902} a^{6}$, $\frac{1}{3781137879198} a^{34} - \frac{796}{1592728677} a^{25} + \frac{18984253}{1592728677} a^{16} - \frac{90745099}{192101706} a^{7}$, $\frac{1}{11343413637594} a^{35} - \frac{796}{4778186031} a^{26} + \frac{18984253}{4778186031} a^{17} + \frac{101356607}{576305118} a^{8}$

Class group and class number

Not computed

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{2108}{2593373031} a^{28} - \frac{31}{4369626} a^{19} - \frac{15467}{4369626} a^{10} + \frac{203391}{87838} a \) (order $54$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_2\times C_{18}$ (as 36T2):

An abelian group of order 36

The 36 conjugacy class representatives for $C_2\times C_{18}$

Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{9})\), 6.0.8732691.1, 6.6.26198073.1, \(\Q(\zeta_{27})^+\), 12.0.686339028913329.1, \(\Q(\zeta_{27})\), 18.0.2322038274967832964613417227771.2, 18.18.6966114824903498893840251683313.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$18^{2}$	R	$18^{2}$	$18^{2}$	R	$18^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/31.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$	$18^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
3	Data not computed
11	Data not computed