Global number field 36.36.2987052991985699215206951151365900403178222386352058024870316677.1: \(\Q(\zeta_{111})^+\)

• Label: 36.36.2987052991...6677.1
• Degree: $36$
• Signature: $[36, 0]$
• Discriminant: $3^{18}\cdot 37^{35}$
• Root discriminant: $57.97$
• Ramified primes: $3, 37$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{36}$ (as 36T1)

Normalized defining polynomial

\( x^{36} - x^{35} - 36 x^{34} + 36 x^{33} + 593 x^{32} - 593 x^{31} - 5919 x^{30} + 5919 x^{29} + 39961 x^{28} - 39961 x^{27} - 192880 x^{26} + 192880 x^{25} + 685907 x^{24} - 685907 x^{23} - 1824913 x^{22} + 1824913 x^{21} + 3651272 x^{20} - 3651272 x^{19} - 5475703 x^{18} + 5475703 x^{17} + 6085132 x^{16} - 6085132 x^{15} - 4909788 x^{14} + 4909788 x^{13} + 2786656 x^{12} - 2786656 x^{11} - 1061566 x^{10} + 1061566 x^{9} + 253044 x^{8} - 253044 x^{7} - 33780 x^{6} + 33780 x^{5} + 2073 x^{4} - 2073 x^{3} - 36 x^{2} + 36 x + 1 \)

Invariants

Degree:		$36$
Signature:		$[36, 0]$
Discriminant:		\(2987052991985699215206951151365900403178222386352058024870316677=3^{18}\cdot 37^{35}\)
Root discriminant:		$57.97$
Ramified primes:		$3, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(111=3\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(2,·)$, $\chi_{111}(4,·)$, $\chi_{111}(5,·)$, $\chi_{111}(7,·)$, $\chi_{111}(8,·)$, $\chi_{111}(10,·)$, $\chi_{111}(14,·)$, $\chi_{111}(16,·)$, $\chi_{111}(17,·)$, $\chi_{111}(20,·)$, $\chi_{111}(23,·)$, $\chi_{111}(25,·)$, $\chi_{111}(28,·)$, $\chi_{111}(29,·)$, $\chi_{111}(32,·)$, $\chi_{111}(34,·)$, $\chi_{111}(35,·)$, $\chi_{111}(40,·)$, $\chi_{111}(46,·)$, $\chi_{111}(49,·)$, $\chi_{111}(50,·)$, $\chi_{111}(56,·)$, $\chi_{111}(58,·)$, $\chi_{111}(59,·)$, $\chi_{111}(64,·)$, $\chi_{111}(67,·)$, $\chi_{111}(68,·)$, $\chi_{111}(70,·)$, $\chi_{111}(73,·)$, $\chi_{111}(80,·)$, $\chi_{111}(85,·)$, $\chi_{111}(89,·)$, $\chi_{111}(92,·)$, $\chi_{111}(98,·)$, $\chi_{111}(100,·)$$\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:		\( 181157165602858830000 \) (assuming GRH)

Galois group

$C_{36}$ (as 36T1):

A cyclic group of order 36

The 36 conjugacy class representatives for $C_{36}$

Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.4.455877.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.129701946277226641077.1, \(\Q(\zeta_{37})^+\)

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	$36$	R	$36$	${\href{/LocalNumberField/7.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$	$36$	$36$	$36$	${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$	R	${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$	${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$	$18^{2}$	$36$

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
37	Data not computed