Global number field 36.0.4579626957516085526487220638656255445999152174466832174789165056.1

• Label: 36.0.45796269575...5056.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $2^{72}\cdot 3^{88}$
• Root discriminant: $58.66$
• Ramified primes: $2, 3$
• Class number: $6327$ (GRH)
• Class group: $[6327]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} + 54 x^{32} + 1143 x^{28} + 12006 x^{24} + 65367 x^{20} + 175806 x^{16} + 203838 x^{12} + 70632 x^{8} + 5481 x^{4} + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
Discriminant:		\(4579626957516085526487220638656255445999152174466832174789165056=2^{72}\cdot 3^{88}\)
Root discriminant:		$58.66$
Ramified primes:		$2, 3$
This field is Galois and abelian over $\Q$.
Conductor:		\(216=2^{3}\cdot 3^{3}\)
Dirichlet character group:		$\lbrace$$\chi_{216}(1,·)$, $\chi_{216}(133,·)$, $\chi_{216}(7,·)$, $\chi_{216}(139,·)$, $\chi_{216}(13,·)$, $\chi_{216}(145,·)$, $\chi_{216}(19,·)$, $\chi_{216}(151,·)$, $\chi_{216}(25,·)$, $\chi_{216}(157,·)$, $\chi_{216}(31,·)$, $\chi_{216}(163,·)$, $\chi_{216}(37,·)$, $\chi_{216}(169,·)$, $\chi_{216}(43,·)$, $\chi_{216}(175,·)$, $\chi_{216}(49,·)$, $\chi_{216}(181,·)$, $\chi_{216}(55,·)$, $\chi_{216}(187,·)$, $\chi_{216}(61,·)$, $\chi_{216}(193,·)$, $\chi_{216}(67,·)$, $\chi_{216}(199,·)$, $\chi_{216}(73,·)$, $\chi_{216}(205,·)$, $\chi_{216}(79,·)$, $\chi_{216}(211,·)$, $\chi_{216}(85,·)$, $\chi_{216}(91,·)$, $\chi_{216}(97,·)$, $\chi_{216}(103,·)$, $\chi_{216}(109,·)$, $\chi_{216}(115,·)$, $\chi_{216}(121,·)$, $\chi_{216}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{11543228840026773577} a^{32} + \frac{5192013369693317289}{11543228840026773577} a^{28} + \frac{2124378184069870956}{11543228840026773577} a^{24} + \frac{3532654901869503285}{11543228840026773577} a^{20} + \frac{182151398991803673}{11543228840026773577} a^{16} + \frac{2574207142871799287}{11543228840026773577} a^{12} + \frac{2950273414691718692}{11543228840026773577} a^{8} - \frac{4022655650947927519}{11543228840026773577} a^{4} - \frac{2446071674144177440}{11543228840026773577}$, $\frac{1}{11543228840026773577} a^{33} + \frac{5192013369693317289}{11543228840026773577} a^{29} + \frac{2124378184069870956}{11543228840026773577} a^{25} + \frac{3532654901869503285}{11543228840026773577} a^{21} + \frac{182151398991803673}{11543228840026773577} a^{17} + \frac{2574207142871799287}{11543228840026773577} a^{13} + \frac{2950273414691718692}{11543228840026773577} a^{9} - \frac{4022655650947927519}{11543228840026773577} a^{5} - \frac{2446071674144177440}{11543228840026773577} a$, $\frac{1}{11543228840026773577} a^{34} + \frac{5192013369693317289}{11543228840026773577} a^{30} + \frac{2124378184069870956}{11543228840026773577} a^{26} + \frac{3532654901869503285}{11543228840026773577} a^{22} + \frac{182151398991803673}{11543228840026773577} a^{18} + \frac{2574207142871799287}{11543228840026773577} a^{14} + \frac{2950273414691718692}{11543228840026773577} a^{10} - \frac{4022655650947927519}{11543228840026773577} a^{6} - \frac{2446071674144177440}{11543228840026773577} a^{2}$, $\frac{1}{11543228840026773577} a^{35} + \frac{5192013369693317289}{11543228840026773577} a^{31} + \frac{2124378184069870956}{11543228840026773577} a^{27} + \frac{3532654901869503285}{11543228840026773577} a^{23} + \frac{182151398991803673}{11543228840026773577} a^{19} + \frac{2574207142871799287}{11543228840026773577} a^{15} + \frac{2950273414691718692}{11543228840026773577} a^{11} - \frac{4022655650947927519}{11543228840026773577} a^{7} - \frac{2446071674144177440}{11543228840026773577} a^{3}$

Class group and class number

$C_{6327}$, which has order $6327$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -\frac{1325536515129434490}{11543228840026773577} a^{35} - \frac{71581407819880161132}{11543228840026773577} a^{31} - \frac{1515219656144029615117}{11543228840026773577} a^{27} - \frac{15917168575556794861326}{11543228840026773577} a^{23} - \frac{86675424330935262178917}{11543228840026773577} a^{19} - \frac{233194425355002444858162}{11543228840026773577} a^{15} - \frac{270608267870879096914902}{11543228840026773577} a^{11} - \frac{94066799521966279280748}{11543228840026773577} a^{7} - \frac{7357328477125103496285}{11543228840026773577} a^{3} \) (order $8$)
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:		\( 44130898078069.0 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{8})\), 6.6.3359232.1, 6.0.419904.1, 6.0.3359232.1, \(\Q(\zeta_{27})^+\), 12.0.722204136308736.1, 18.18.132173713091594538512566714368.1, 18.0.258151783382020583032356864.7, 18.0.132173713091594538512566714368.2

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	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$	$18^{2}$

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