Global number field 36.0.392893567271872510941083606170645734076396278452324169894854656.1

• Label: 36.0.39289356727...4656.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $2^{72}\cdot 19^{32}$
• Root discriminant: $54.79$
• Ramified primes: $2, 19$
• Class number: $3249$ (GRH)
• Class group: $[19, 171]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} + 49 x^{32} + 932 x^{28} + 8695 x^{24} + 41461 x^{20} + 96055 x^{16} + 93536 x^{12} + 28314 x^{8} + 1365 x^{4} + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(392893567271872510941083606170645734076396278452324169894854656=2^{72}\cdot 19^{32}\)
Root discriminant:		$54.79$
Ramified primes:		$2, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(152=2^{3}\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{152}(1,·)$, $\chi_{152}(131,·)$, $\chi_{152}(5,·)$, $\chi_{152}(7,·)$, $\chi_{152}(9,·)$, $\chi_{152}(11,·)$, $\chi_{152}(17,·)$, $\chi_{152}(149,·)$, $\chi_{152}(23,·)$, $\chi_{152}(25,·)$, $\chi_{152}(35,·)$, $\chi_{152}(39,·)$, $\chi_{152}(43,·)$, $\chi_{152}(45,·)$, $\chi_{152}(47,·)$, $\chi_{152}(49,·)$, $\chi_{152}(137,·)$, $\chi_{152}(61,·)$, $\chi_{152}(63,·)$, $\chi_{152}(139,·)$, $\chi_{152}(73,·)$, $\chi_{152}(55,·)$, $\chi_{152}(77,·)$, $\chi_{152}(81,·)$, $\chi_{152}(83,·)$, $\chi_{152}(85,·)$, $\chi_{152}(87,·)$, $\chi_{152}(93,·)$, $\chi_{152}(99,·)$, $\chi_{152}(101,·)$, $\chi_{152}(111,·)$, $\chi_{152}(115,·)$, $\chi_{152}(119,·)$, $\chi_{152}(121,·)$, $\chi_{152}(123,·)$, $\chi_{152}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{37} a^{28} + \frac{16}{37} a^{24} + \frac{11}{37} a^{16} - \frac{9}{37} a^{12} + \frac{10}{37} a^{4} + \frac{12}{37}$, $\frac{1}{37} a^{29} + \frac{16}{37} a^{25} + \frac{11}{37} a^{17} - \frac{9}{37} a^{13} + \frac{10}{37} a^{5} + \frac{12}{37} a$, $\frac{1}{37} a^{30} + \frac{16}{37} a^{26} + \frac{11}{37} a^{18} - \frac{9}{37} a^{14} + \frac{10}{37} a^{6} + \frac{12}{37} a^{2}$, $\frac{1}{37} a^{31} + \frac{16}{37} a^{27} + \frac{11}{37} a^{19} - \frac{9}{37} a^{15} + \frac{10}{37} a^{7} + \frac{12}{37} a^{3}$, $\frac{1}{27985118605791989} a^{32} - \frac{11925490864898}{27985118605791989} a^{28} + \frac{6493925443472047}{27985118605791989} a^{24} - \frac{5571468246550776}{27985118605791989} a^{20} + \frac{11480800345475487}{27985118605791989} a^{16} - \frac{8547611653942815}{27985118605791989} a^{12} + \frac{10111138899704072}{27985118605791989} a^{8} - \frac{5050809295317656}{27985118605791989} a^{4} + \frac{1260169341003405}{27985118605791989}$, $\frac{1}{27985118605791989} a^{33} - \frac{11925490864898}{27985118605791989} a^{29} + \frac{6493925443472047}{27985118605791989} a^{25} - \frac{5571468246550776}{27985118605791989} a^{21} + \frac{11480800345475487}{27985118605791989} a^{17} - \frac{8547611653942815}{27985118605791989} a^{13} + \frac{10111138899704072}{27985118605791989} a^{9} - \frac{5050809295317656}{27985118605791989} a^{5} + \frac{1260169341003405}{27985118605791989} a$, $\frac{1}{27985118605791989} a^{34} - \frac{11925490864898}{27985118605791989} a^{30} + \frac{6493925443472047}{27985118605791989} a^{26} - \frac{5571468246550776}{27985118605791989} a^{22} + \frac{11480800345475487}{27985118605791989} a^{18} - \frac{8547611653942815}{27985118605791989} a^{14} + \frac{10111138899704072}{27985118605791989} a^{10} - \frac{5050809295317656}{27985118605791989} a^{6} + \frac{1260169341003405}{27985118605791989} a^{2}$, $\frac{1}{27985118605791989} a^{35} - \frac{11925490864898}{27985118605791989} a^{31} + \frac{6493925443472047}{27985118605791989} a^{27} - \frac{5571468246550776}{27985118605791989} a^{23} + \frac{11480800345475487}{27985118605791989} a^{19} - \frac{8547611653942815}{27985118605791989} a^{15} + \frac{10111138899704072}{27985118605791989} a^{11} - \frac{5050809295317656}{27985118605791989} a^{7} + \frac{1260169341003405}{27985118605791989} a^{3}$

Class group and class number

$C_{19}\times C_{171}$, which has order $3249$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( \frac{4933715287540570}{27985118605791989} a^{35} + \frac{241728563142432074}{27985118605791989} a^{31} + \frac{4597078388635635768}{27985118605791989} a^{27} + \frac{42877073860924213654}{27985118605791989} a^{23} + \frac{204358106636401389981}{27985118605791989} a^{19} + \frac{472982025320174328996}{27985118605791989} a^{15} + \frac{459425171055352514653}{27985118605791989} a^{11} + \frac{137874115120972430578}{27985118605791989} a^{7} + \frac{6314209051969526189}{27985118605791989} a^{3} \) (order $8$)
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:		\( 28122649019657.055 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 3.3.361.1, \(\Q(\zeta_{8})\), 6.0.8340544.1, 6.6.66724352.1, 6.0.66724352.1, \(\Q(\zeta_{19})^+\), 12.0.284936905588473856.1, 18.0.75613185918270483380568064.1, 18.18.38713951190154487490850848768.1, 18.0.38713951190154487490850848768.1

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	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$	$18^{2}$	${\href{/LocalNumberField/17.9.0.1}{9} }^{4}$	R	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$	${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$

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