Global number field 36.36.799622233646074762983150698451178476894456963777140963130998784.1: \(\Q(\zeta_{228})^+\)

• Label: 36.36.7996222336...8784.1
• Degree: $36$
• Signature: $[36, 0]$
• Discriminant: $2^{36}\cdot 3^{18}\cdot 19^{34}$
• Root discriminant: $55.89$
• Ramified primes: $2, 3, 19$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} - 35 x^{34} + 560 x^{32} - 5426 x^{30} + 35554 x^{28} - 166634 x^{26} + 576201 x^{24} - 1494747 x^{22} + 2929464 x^{20} - 4334718 x^{18} + 4805781 x^{16} - 3932287 x^{14} + 2318239 x^{12} - 949739 x^{10} + 256105 x^{8} - 41769 x^{6} + 3570 x^{4} - 120 x^{2} + 1 \)

Invariants

Degree:		$36$
Signature:		$[36, 0]$
Discriminant:		\(799622233646074762983150698451178476894456963777140963130998784=2^{36}\cdot 3^{18}\cdot 19^{34}\)
Root discriminant:		$55.89$
Ramified primes:		$2, 3, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(228=2^{2}\cdot 3\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{228}(1,·)$, $\chi_{228}(131,·)$, $\chi_{228}(11,·)$, $\chi_{228}(23,·)$, $\chi_{228}(25,·)$, $\chi_{228}(29,·)$, $\chi_{228}(31,·)$, $\chi_{228}(35,·)$, $\chi_{228}(41,·)$, $\chi_{228}(173,·)$, $\chi_{228}(47,·)$, $\chi_{228}(49,·)$, $\chi_{228}(53,·)$, $\chi_{228}(185,·)$, $\chi_{228}(151,·)$, $\chi_{228}(61,·)$, $\chi_{228}(191,·)$, $\chi_{228}(65,·)$, $\chi_{228}(67,·)$, $\chi_{228}(73,·)$, $\chi_{228}(119,·)$, $\chi_{228}(79,·)$, $\chi_{228}(83,·)$, $\chi_{228}(85,·)$, $\chi_{228}(215,·)$, $\chi_{228}(89,·)$, $\chi_{228}(91,·)$, $\chi_{228}(221,·)$, $\chi_{228}(223,·)$, $\chi_{228}(103,·)$, $\chi_{228}(157,·)$, $\chi_{228}(113,·)$, $\chi_{228}(211,·)$, $\chi_{228}(169,·)$, $\chi_{228}(121,·)$, $\chi_{228}(127,·)$$\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:		\( 104443369202940970000 \) (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, \(\Q(\sqrt{3}, \sqrt{19})\), 6.6.225194688.1, 6.6.158470336.1, 6.6.66854673.1, \(\Q(\zeta_{19})^+\), 12.12.18307265748733661184.1, 18.18.1488294338429317924379721203712.1, \(\Q(\zeta_{76})^+\), \(\Q(\zeta_{57})^+\)

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	R	R	$18^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$	$18^{2}$	$18^{2}$	R	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$	$18^{2}$	$18^{2}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
2	Data not computed
3	Data not computed
19	Data not computed