Global number field 36.0.599781089369859106058502013153430001897393515831230464.1: \(\Q(\zeta_{108})\)

• Label: 36.0.59978108936...0464.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $2^{36}\cdot 3^{90}$
• Root discriminant: $31.18$
• Ramified primes: $2, 3$
• Class number: $19$ (GRH)
• Class group: $[19]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - x^{18} + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(599781089369859106058502013153430001897393515831230464=2^{36}\cdot 3^{90}\)
Root discriminant:		$31.18$
Ramified primes:		$2, 3$
This field is Galois and abelian over $\Q$.
Conductor:		\(108=2^{2}\cdot 3^{3}\)
Dirichlet character group:		$\lbrace$$\chi_{108}(1,·)$, $\chi_{108}(5,·)$, $\chi_{108}(7,·)$, $\chi_{108}(11,·)$, $\chi_{108}(13,·)$, $\chi_{108}(17,·)$, $\chi_{108}(19,·)$, $\chi_{108}(23,·)$, $\chi_{108}(25,·)$, $\chi_{108}(29,·)$, $\chi_{108}(31,·)$, $\chi_{108}(35,·)$, $\chi_{108}(37,·)$, $\chi_{108}(41,·)$, $\chi_{108}(43,·)$, $\chi_{108}(47,·)$, $\chi_{108}(49,·)$, $\chi_{108}(53,·)$, $\chi_{108}(55,·)$, $\chi_{108}(59,·)$, $\chi_{108}(61,·)$, $\chi_{108}(65,·)$, $\chi_{108}(67,·)$, $\chi_{108}(71,·)$, $\chi_{108}(73,·)$, $\chi_{108}(77,·)$, $\chi_{108}(79,·)$, $\chi_{108}(83,·)$, $\chi_{108}(85,·)$, $\chi_{108}(89,·)$, $\chi_{108}(91,·)$, $\chi_{108}(95,·)$, $\chi_{108}(97,·)$, $\chi_{108}(101,·)$, $\chi_{108}(103,·)$, $\chi_{108}(107,·)$$\rbrace$
This is a CM field.

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

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

Class group and class number

$C_{19}$, which has order $19$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( a \) (order $108$)
Fundamental units:		\( a^{26} - a^{8} - a^{2} \),  \( a^{32} - a^{20} + a^{2} \),  \( a^{4} + 1 \),  \( a^{34} - a^{16} + a^{12} \),  \( a^{28} + a^{20} - a^{10} - a^{2} \),  \( a^{10} - 1 \),  \( a^{32} + a^{16} \),  \( a^{34} - 1 \),  \( a^{29} - a^{20} \),  \( a^{15} - 1 \),  \( a^{34} - a^{31} - a^{16} + a^{13} \),  \( a^{5} - 1 \),  \( a^{13} - 1 \),  \( a - 1 \),  \( a^{17} - 1 \),  \( a^{11} - 1 \),  \( a^{7} - 1 \) (assuming GRH)
Regulator:		\( 2861978396449.9033 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), 6.0.419904.1, \(\Q(\zeta_{9})\), \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{27})^+\), \(\Q(\zeta_{36})\), 18.0.258151783382020583032356864.7, \(\Q(\zeta_{27})\), \(\Q(\zeta_{108})^+\)

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	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/13.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$	$18^{2}$

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
2	Data not computed
3	Data not computed