Global number field 36.36.529834441956838404754859356629890081471398054377951743656484405248.1: \(\Q(\zeta_{148})^+\)

• Label: 36.36.5298344419...5248.1
• Degree: $36$
• Signature: $[36, 0]$
• Discriminant: $2^{36}\cdot 37^{35}$
• Root discriminant: $66.94$
• Ramified primes: $2, 37$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{36}$ (as 36T1)

Normalized defining polynomial

\( x^{36} - 37 x^{34} + 629 x^{32} - 6512 x^{30} + 45880 x^{28} - 232841 x^{26} + 878787 x^{24} - 2510820 x^{22} + 5476185 x^{20} - 9126975 x^{18} + 11560835 x^{16} - 10994920 x^{14} + 7696444 x^{12} - 3848222 x^{10} + 1314610 x^{8} - 286824 x^{6} + 35853 x^{4} - 2109 x^{2} + 37 \)

Invariants

Degree:		$36$
Signature:		$[36, 0]$
Discriminant:		\(529834441956838404754859356629890081471398054377951743656484405248=2^{36}\cdot 37^{35}\)
Root discriminant:		$66.94$
Ramified primes:		$2, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(148=2^{2}\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{148}(1,·)$, $\chi_{148}(131,·)$, $\chi_{148}(135,·)$, $\chi_{148}(9,·)$, $\chi_{148}(141,·)$, $\chi_{148}(15,·)$, $\chi_{148}(145,·)$, $\chi_{148}(19,·)$, $\chi_{148}(21,·)$, $\chi_{148}(23,·)$, $\chi_{148}(25,·)$, $\chi_{148}(31,·)$, $\chi_{148}(33,·)$, $\chi_{148}(35,·)$, $\chi_{148}(39,·)$, $\chi_{148}(41,·)$, $\chi_{148}(43,·)$, $\chi_{148}(49,·)$, $\chi_{148}(91,·)$, $\chi_{148}(51,·)$, $\chi_{148}(53,·)$, $\chi_{148}(137,·)$, $\chi_{148}(59,·)$, $\chi_{148}(65,·)$, $\chi_{148}(73,·)$, $\chi_{148}(55,·)$, $\chi_{148}(77,·)$, $\chi_{148}(79,·)$, $\chi_{148}(81,·)$, $\chi_{148}(85,·)$, $\chi_{148}(87,·)$, $\chi_{148}(143,·)$, $\chi_{148}(101,·)$, $\chi_{148}(103,·)$, $\chi_{148}(119,·)$, $\chi_{148}(121,·)$$\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:		\( 2745123952948895000000 \) (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, 4.4.810448.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.728750578808669851648.1, \(\Q(\zeta_{37})^+\)

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	R	${\href{/LocalNumberField/3.9.0.1}{9} }^{4}$	$36$	$18^{2}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$	$36$	$36$	$36$	${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$	R	$18^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$	${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$	$36$

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
2	Data not computed
37	Data not computed