Global number field 36.0.11636034958735032166924075841251447518799351583251569.1: \(\Q(\zeta_{57})\)

• Label: 36.0.11636034958...1569.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $3^{18}\cdot 19^{34}$
• Root discriminant: $27.94$
• Ramified primes: $3, 19$
• Class number: $9$ (GRH)
• Class group: $[9]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - x^{35} + x^{33} - x^{32} + x^{30} - x^{29} + x^{27} - x^{26} + x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{16} + x^{15} - x^{13} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(11636034958735032166924075841251447518799351583251569=3^{18}\cdot 19^{34}\)
Root discriminant:		$27.94$
Ramified primes:		$3, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(57=3\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{57}(1,·)$, $\chi_{57}(2,·)$, $\chi_{57}(4,·)$, $\chi_{57}(5,·)$, $\chi_{57}(7,·)$, $\chi_{57}(8,·)$, $\chi_{57}(10,·)$, $\chi_{57}(11,·)$, $\chi_{57}(13,·)$, $\chi_{57}(14,·)$, $\chi_{57}(16,·)$, $\chi_{57}(17,·)$, $\chi_{57}(20,·)$, $\chi_{57}(22,·)$, $\chi_{57}(23,·)$, $\chi_{57}(25,·)$, $\chi_{57}(26,·)$, $\chi_{57}(28,·)$, $\chi_{57}(29,·)$, $\chi_{57}(31,·)$, $\chi_{57}(32,·)$, $\chi_{57}(34,·)$, $\chi_{57}(35,·)$, $\chi_{57}(37,·)$, $\chi_{57}(40,·)$, $\chi_{57}(41,·)$, $\chi_{57}(43,·)$, $\chi_{57}(44,·)$, $\chi_{57}(46,·)$, $\chi_{57}(47,·)$, $\chi_{57}(49,·)$, $\chi_{57}(50,·)$, $\chi_{57}(52,·)$, $\chi_{57}(53,·)$, $\chi_{57}(55,·)$, $\chi_{57}(56,·)$$\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

$C_{9}$, which has order $9$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -a \) (order $114$)
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:		\( 719116485989.9799 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{57}) \), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{-19})\), 6.0.3518667.1, 6.0.2476099.1, 6.6.66854673.1, \(\Q(\zeta_{19})^+\), 12.0.4469547301936929.1, 18.0.5677392343251487443465123.1, \(\Q(\zeta_{19})\), \(\Q(\zeta_{57})^+\)

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	$18^{2}$	R	$18^{2}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{12}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	R	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$	$18^{2}$	${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$	$18^{2}$	$18^{2}$	$18^{2}$

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
3	Data not computed
19	Data not computed