Global number field 36.36.41216642617644769738384985747906299013992369570201489573102485504.1: \(\Q(\zeta_{216})^+\)

• Label: 36.36.4121664261...5504.1
• Degree: $36$
• Signature: $[36, 0]$
• Discriminant: $2^{72}\cdot 3^{90}$
• Root discriminant: $62.35$
• Ramified primes: $2, 3$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - 36 x^{34} + 594 x^{32} - 5952 x^{30} + 40455 x^{28} - 197316 x^{26} + 712530 x^{24} - 1937520 x^{22} + 3996135 x^{20} - 6249100 x^{18} + 7354710 x^{16} - 6418656 x^{14} + 4056234 x^{12} - 1790712 x^{10} + 523260 x^{8} - 93024 x^{6} + 8721 x^{4} - 324 x^{2} + 1 \)

Invariants

Degree:		$36$
Signature:		$[36, 0]$
Discriminant:		\(41216642617644769738384985747906299013992369570201489573102485504=2^{72}\cdot 3^{90}\)
Root discriminant:		$62.35$
Ramified primes:		$2, 3$
This field is Galois and abelian over $\Q$.
Conductor:		\(216=2^{3}\cdot 3^{3}\)
Dirichlet character group:		$\lbrace$$\chi_{216}(1,·)$, $\chi_{216}(131,·)$, $\chi_{216}(133,·)$, $\chi_{216}(11,·)$, $\chi_{216}(13,·)$, $\chi_{216}(143,·)$, $\chi_{216}(145,·)$, $\chi_{216}(23,·)$, $\chi_{216}(25,·)$, $\chi_{216}(155,·)$, $\chi_{216}(157,·)$, $\chi_{216}(35,·)$, $\chi_{216}(37,·)$, $\chi_{216}(167,·)$, $\chi_{216}(169,·)$, $\chi_{216}(47,·)$, $\chi_{216}(49,·)$, $\chi_{216}(179,·)$, $\chi_{216}(181,·)$, $\chi_{216}(59,·)$, $\chi_{216}(61,·)$, $\chi_{216}(191,·)$, $\chi_{216}(193,·)$, $\chi_{216}(71,·)$, $\chi_{216}(73,·)$, $\chi_{216}(203,·)$, $\chi_{216}(205,·)$, $\chi_{216}(83,·)$, $\chi_{216}(85,·)$, $\chi_{216}(215,·)$, $\chi_{216}(95,·)$, $\chi_{216}(97,·)$, $\chi_{216}(107,·)$, $\chi_{216}(109,·)$, $\chi_{216}(119,·)$, $\chi_{216}(121,·)$$\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:		\( 682049432255791000000 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{2}, \sqrt{3})\), 6.6.3359232.1, 6.6.10077696.1, \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{27})^+\), \(\Q(\zeta_{72})^+\), 18.18.132173713091594538512566714368.1, 18.18.396521139274783615537700143104.1, \(\Q(\zeta_{108})^+\)

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	R	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$	$18^{2}$

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
2	Data not computed
3	Data not computed