Global number field 36.36.223775506846460533290697977531706040570972271263599395751953125.1: \(\Q(\zeta_{95})^+\)

• Label: 36.36.2237755068...3125.1
• Degree: $36$
• Signature: $[36, 0]$
• Discriminant: $5^{27}\cdot 19^{34}$
• Root discriminant: $53.94$
• Ramified primes: $5, 19$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{36}$ (as 36T1)

Normalized defining polynomial

\( x^{36} - x^{35} - 36 x^{34} + 35 x^{33} + 594 x^{32} - 559 x^{31} - 5953 x^{30} + 5394 x^{29} + 40485 x^{28} - 35091 x^{27} - 197720 x^{26} + 162629 x^{25} + 715754 x^{24} - 553125 x^{23} - 1954471 x^{22} + 1401346 x^{21} + 4057888 x^{20} - 2656542 x^{19} - 6408680 x^{18} + 3752139 x^{17} + 7649116 x^{16} - 3896996 x^{15} - 6803518 x^{14} + 2906674 x^{13} + 4404908 x^{12} - 1498899 x^{11} - 2000649 x^{10} + 503479 x^{9} + 601194 x^{8} - 100432 x^{7} - 108458 x^{6} + 10534 x^{5} + 9885 x^{4} - 605 x^{3} - 340 x^{2} + 20 x + 1 \)

Invariants

Degree:		$36$
Signature:		$[36, 0]$
Discriminant:		\(223775506846460533290697977531706040570972271263599395751953125=5^{27}\cdot 19^{34}\)
Root discriminant:		$53.94$
Ramified primes:		$5, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(95=5\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{95}(1,·)$, $\chi_{95}(2,·)$, $\chi_{95}(3,·)$, $\chi_{95}(4,·)$, $\chi_{95}(6,·)$, $\chi_{95}(8,·)$, $\chi_{95}(9,·)$, $\chi_{95}(11,·)$, $\chi_{95}(12,·)$, $\chi_{95}(13,·)$, $\chi_{95}(16,·)$, $\chi_{95}(18,·)$, $\chi_{95}(22,·)$, $\chi_{95}(24,·)$, $\chi_{95}(26,·)$, $\chi_{95}(27,·)$, $\chi_{95}(32,·)$, $\chi_{95}(33,·)$, $\chi_{95}(36,·)$, $\chi_{95}(37,·)$, $\chi_{95}(39,·)$, $\chi_{95}(44,·)$, $\chi_{95}(48,·)$, $\chi_{95}(49,·)$, $\chi_{95}(52,·)$, $\chi_{95}(53,·)$, $\chi_{95}(54,·)$, $\chi_{95}(61,·)$, $\chi_{95}(64,·)$, $\chi_{95}(66,·)$, $\chi_{95}(67,·)$, $\chi_{95}(72,·)$, $\chi_{95}(74,·)$, $\chi_{95}(78,·)$, $\chi_{95}(81,·)$, $\chi_{95}(88,·)$$\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:		\( 58714175232406490000 \) (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

\(\Q(\sqrt{5}) \), 3.3.361.1, 4.4.45125.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.12.11974738784767578125.1, 18.18.563362135874260093126953125.1

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	$36$	$36$	R	${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$	$36$	$36$	R	$36$	${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{9}$	$18^{2}$	$36$	$36$	$36$	${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
5	Data not computed
19	Data not computed