Global number field 36.0.799622233646074762983150698451178476894456963777140963130998784.1

• Label: 36.0.79962223364...8784.1
• Degree: $36$
• Signature: $[0, 18]$
• Discriminant: $2^{36}\cdot 3^{18}\cdot 19^{34}$
• Root discriminant: $55.89$
• Ramified primes: $2, 3, 19$
• Class number: $52934$ (GRH)
• Class group: $[52934]$ (GRH)
• Galois group: $C_2\times C_{18}$ (as 36T2)

Normalized defining polynomial

\( x^{36} + 37 x^{34} + 628 x^{32} + 6478 x^{30} + 45354 x^{28} + 227942 x^{26} + 848161 x^{24} + 2375189 x^{22} + 5038516 x^{20} + 8084594 x^{18} + 9725341 x^{16} + 8624289 x^{14} + 5490811 x^{12} + 2413645 x^{10} + 691677 x^{8} + 118407 x^{6} + 10470 x^{4} + 360 x^{2} + 1 \)

Invariants

Degree:		$36$
Signature:		$[0, 18]$
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$$\chi_{228}(1,·)$, $\chi_{228}(7,·)$, $\chi_{228}(139,·)$, $\chi_{228}(143,·)$, $\chi_{228}(25,·)$, $\chi_{228}(155,·)$, $\chi_{228}(29,·)$, $\chi_{228}(163,·)$, $\chi_{228}(167,·)$, $\chi_{228}(41,·)$, $\chi_{228}(43,·)$, $\chi_{228}(173,·)$, $\chi_{228}(175,·)$, $\chi_{228}(49,·)$, $\chi_{228}(179,·)$, $\chi_{228}(53,·)$, $\chi_{228}(55,·)$, $\chi_{228}(185,·)$, $\chi_{228}(59,·)$, $\chi_{228}(61,·)$, $\chi_{228}(65,·)$, $\chi_{228}(71,·)$, $\chi_{228}(73,·)$, $\chi_{228}(203,·)$, $\chi_{228}(85,·)$, $\chi_{228}(89,·)$, $\chi_{228}(221,·)$, $\chi_{228}(199,·)$, $\chi_{228}(227,·)$, $\chi_{228}(107,·)$, $\chi_{228}(157,·)$, $\chi_{228}(113,·)$, $\chi_{228}(115,·)$, $\chi_{228}(187,·)$, $\chi_{228}(169,·)$, $\chi_{228}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$

Class group and class number

$C_{52934}$, which has order $52934$ (assuming GRH)

Unit group

Rank:		$17$
Torsion generator:		\( -a^{19} - 19 a^{17} - 152 a^{15} - 665 a^{13} - 1729 a^{11} - 2717 a^{9} - 2508 a^{7} - 1254 a^{5} - 285 a^{3} - 19 a \) (order $4$)
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:		\( 1438232971979.9597 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{57}) \), 3.3.361.1, \(\Q(i, \sqrt{57})\), 6.0.8340544.1, 6.0.4278699072.1, 6.6.66854673.1, \(\Q(\zeta_{19})^+\), 12.0.18307265748733661184.1, 18.0.75613185918270483380568064.1, 18.0.28277592430157040563214702870528.1, \(\Q(\zeta_{57})^+\)

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	$18^{2}$	${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$	${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$	$18^{2}$	$18^{2}$	${\href{/LocalNumberField/53.9.0.1}{9} }^{4}$	$18^{2}$

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
19	Data not computed