Global number field 36.0.141834577785145976449731181827603110001579056521289025332042530816.2

• Label: 36.0.14183457778...0816.2
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $2^{72}\cdot 19^{34}$
• Root discriminant: $64.53$
• Ramified primes: $2, 19$
• Class number: $19494$ (GRH)
• Class group: $[19, 1026]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} + 57 x^{32} + 1292 x^{28} + 14839 x^{24} + 91181 x^{20} + 291327 x^{16} + 436088 x^{12} + 246202 x^{8} + 33573 x^{4} + 361 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(141834577785145976449731181827603110001579056521289025332042530816=2^{72}\cdot 19^{34}\)
Root discriminant:		$64.53$
Ramified primes:		$2, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(152=2^{3}\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{152}(1,·)$, $\chi_{152}(3,·)$, $\chi_{152}(7,·)$, $\chi_{152}(9,·)$, $\chi_{152}(13,·)$, $\chi_{152}(17,·)$, $\chi_{152}(147,·)$, $\chi_{152}(21,·)$, $\chi_{152}(23,·)$, $\chi_{152}(25,·)$, $\chi_{152}(27,·)$, $\chi_{152}(29,·)$, $\chi_{152}(37,·)$, $\chi_{152}(39,·)$, $\chi_{152}(47,·)$, $\chi_{152}(49,·)$, $\chi_{152}(51,·)$, $\chi_{152}(53,·)$, $\chi_{152}(137,·)$, $\chi_{152}(59,·)$, $\chi_{152}(63,·)$, $\chi_{152}(67,·)$, $\chi_{152}(69,·)$, $\chi_{152}(73,·)$, $\chi_{152}(55,·)$, $\chi_{152}(141,·)$, $\chi_{152}(81,·)$, $\chi_{152}(75,·)$, $\chi_{152}(87,·)$, $\chi_{152}(91,·)$, $\chi_{152}(107,·)$, $\chi_{152}(109,·)$, $\chi_{152}(111,·)$, $\chi_{152}(117,·)$, $\chi_{152}(119,·)$, $\chi_{152}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18}$, $\frac{1}{19} a^{19}$, $\frac{1}{19} a^{20}$, $\frac{1}{19} a^{21}$, $\frac{1}{19} a^{22}$, $\frac{1}{19} a^{23}$, $\frac{1}{19} a^{24}$, $\frac{1}{19} a^{25}$, $\frac{1}{19} a^{26}$, $\frac{1}{19} a^{27}$, $\frac{1}{703} a^{28} + \frac{5}{703} a^{24} - \frac{13}{703} a^{20} + \frac{8}{37} a^{16} - \frac{8}{37} a^{12} - \frac{11}{37} a^{8} - \frac{10}{37} a^{4} - \frac{2}{37}$, $\frac{1}{703} a^{29} + \frac{5}{703} a^{25} - \frac{13}{703} a^{21} + \frac{8}{37} a^{17} - \frac{8}{37} a^{13} - \frac{11}{37} a^{9} - \frac{10}{37} a^{5} - \frac{2}{37} a$, $\frac{1}{703} a^{30} + \frac{5}{703} a^{26} - \frac{13}{703} a^{22} + \frac{4}{703} a^{18} - \frac{8}{37} a^{14} - \frac{11}{37} a^{10} - \frac{10}{37} a^{6} - \frac{2}{37} a^{2}$, $\frac{1}{703} a^{31} + \frac{5}{703} a^{27} - \frac{13}{703} a^{23} + \frac{4}{703} a^{19} - \frac{8}{37} a^{15} - \frac{11}{37} a^{11} - \frac{10}{37} a^{7} - \frac{2}{37} a^{3}$, $\frac{1}{1202291036053231} a^{32} - \frac{360628029338}{1202291036053231} a^{28} + \frac{18071998860342}{1202291036053231} a^{24} + \frac{19263197343064}{1202291036053231} a^{20} + \frac{22374821068296}{63278475581749} a^{16} - \frac{29421940747462}{63278475581749} a^{12} + \frac{6900618736258}{63278475581749} a^{8} + \frac{2796551215897}{63278475581749} a^{4} + \frac{9414536636459}{63278475581749}$, $\frac{1}{1202291036053231} a^{33} - \frac{360628029338}{1202291036053231} a^{29} + \frac{18071998860342}{1202291036053231} a^{25} + \frac{19263197343064}{1202291036053231} a^{21} + \frac{22374821068296}{63278475581749} a^{17} - \frac{29421940747462}{63278475581749} a^{13} + \frac{6900618736258}{63278475581749} a^{9} + \frac{2796551215897}{63278475581749} a^{5} + \frac{9414536636459}{63278475581749} a$, $\frac{1}{1202291036053231} a^{34} - \frac{360628029338}{1202291036053231} a^{30} + \frac{18071998860342}{1202291036053231} a^{26} + \frac{19263197343064}{1202291036053231} a^{22} - \frac{17827728774619}{1202291036053231} a^{18} - \frac{29421940747462}{63278475581749} a^{14} + \frac{6900618736258}{63278475581749} a^{10} + \frac{2796551215897}{63278475581749} a^{6} + \frac{9414536636459}{63278475581749} a^{2}$, $\frac{1}{1202291036053231} a^{35} - \frac{360628029338}{1202291036053231} a^{31} + \frac{18071998860342}{1202291036053231} a^{27} + \frac{19263197343064}{1202291036053231} a^{23} - \frac{17827728774619}{1202291036053231} a^{19} - \frac{29421940747462}{63278475581749} a^{15} + \frac{6900618736258}{63278475581749} a^{11} + \frac{2796551215897}{63278475581749} a^{7} + \frac{9414536636459}{63278475581749} a^{3}$

Class group and class number

$C_{19}\times C_{1026}$, which has order $19494$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{270553808645}{1202291036053231} a^{34} - \frac{15427867616042}{1202291036053231} a^{30} - \frac{349915866100519}{1202291036053231} a^{26} - \frac{4022958387146530}{1202291036053231} a^{22} - \frac{24764481383450907}{1202291036053231} a^{18} - \frac{4179650730230042}{63278475581749} a^{14} - \frac{6312993565535598}{63278475581749} a^{10} - \frac{3675142499306004}{63278475581749} a^{6} - \frac{626229224955623}{63278475581749} a^{2} \) (order $4$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 83538158776577.44 \) (assuming GRH)

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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{38}) \), \(\Q(\sqrt{-38}) \), 3.3.361.1, \(\Q(i, \sqrt{38})\), 6.0.8340544.1, 6.6.1267762688.1, 6.0.1267762688.1, \(\Q(\zeta_{19})^+\), 12.0.102862222917439062016.2, 18.0.75613185918270483380568064.1, 18.18.735565072612935262326166126592.1, 18.0.735565072612935262326166126592.1

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	R	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/13.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$	R	$18^{2}$	${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.1.0.1}{1} }^{36}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$	$18^{2}$

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
2	Data not computed
19	Data not computed