Global number field 36.36.141834577785145976449731181827603110001579056521289025332042530816.1: \(\Q(\zeta_{152})^+\)

• Label: 36.36.1418345777...0816.1
• Degree: $36$
• Signature: $[36, 0]$
• Discriminant: $2^{72}\cdot 19^{34}$
• Root discriminant: $64.53$
• Ramified primes: $2, 19$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - 36 x^{34} + 593 x^{32} - 5920 x^{30} + 39992 x^{28} - 193312 x^{26} + 689479 x^{24} - 1844392 x^{22} + 3724921 x^{20} - 5673268 x^{18} + 6463399 x^{16} - 5422832 x^{14} + 3269436 x^{12} - 1365584 x^{10} + 374154 x^{8} - 61776 x^{6} + 5325 x^{4} - 180 x^{2} + 1 \)

Invariants

Degree:		$36$
Signature:		$[36, 0]$
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$$\chi_{152}(1,·)$, $\chi_{152}(3,·)$, $\chi_{152}(5,·)$, $\chi_{152}(135,·)$, $\chi_{152}(9,·)$, $\chi_{152}(15,·)$, $\chi_{152}(17,·)$, $\chi_{152}(147,·)$, $\chi_{152}(149,·)$, $\chi_{152}(151,·)$, $\chi_{152}(25,·)$, $\chi_{152}(27,·)$, $\chi_{152}(31,·)$, $\chi_{152}(91,·)$, $\chi_{152}(45,·)$, $\chi_{152}(49,·)$, $\chi_{152}(51,·)$, $\chi_{152}(137,·)$, $\chi_{152}(59,·)$, $\chi_{152}(61,·)$, $\chi_{152}(67,·)$, $\chi_{152}(71,·)$, $\chi_{152}(73,·)$, $\chi_{152}(75,·)$, $\chi_{152}(77,·)$, $\chi_{152}(79,·)$, $\chi_{152}(81,·)$, $\chi_{152}(85,·)$, $\chi_{152}(143,·)$, $\chi_{152}(93,·)$, $\chi_{152}(101,·)$, $\chi_{152}(103,·)$, $\chi_{152}(107,·)$, $\chi_{152}(121,·)$, $\chi_{152}(125,·)$, $\chi_{152}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$35$
Torsion generator:		\( -1 \) (order $2$)
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:		\( 1426609831093226200000 \) (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

\(\Q(\sqrt{38}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{2}) \), 3.3.361.1, \(\Q(\sqrt{2}, \sqrt{19})\), 6.6.1267762688.1, 6.6.158470336.1, 6.6.66724352.1, \(\Q(\zeta_{19})^+\), 12.12.102862222917439062016.1, 18.18.735565072612935262326166126592.1, \(\Q(\zeta_{76})^+\), 18.18.38713951190154487490850848768.1

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	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$

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