Global number field 40.40.100383397447978918530459891214693626269465146363712847232818603515625.1: \(\Q(\zeta_{165})^+\)

• Label: 40.40.1003833974...5625.1
• Degree: $40$
• Signature: $[40, 0]$
• Discriminant: $3^{20}\cdot 5^{30}\cdot 11^{36}$
• Root discriminant: $50.12$
• Ramified primes: $3, 5, 11$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} - x^{39} - 39 x^{38} + 39 x^{37} + 702 x^{36} - 701 x^{35} - 7736 x^{34} + 7701 x^{33} + 58378 x^{32} - 57817 x^{31} - 319703 x^{30} + 314247 x^{29} + 1313873 x^{28} - 1277913 x^{27} - 4133168 x^{26} + 3963256 x^{25} + 10063407 x^{24} - 9469608 x^{23} - 19054236 x^{22} + 17493203 x^{21} + 28043254 x^{20} - 24933608 x^{19} - 31915288 x^{18} + 27222459 x^{17} + 27803858 x^{16} - 22476466 x^{15} - 18249278 x^{14} + 13764066 x^{13} + 8815690 x^{12} - 6077890 x^{11} - 3029168 x^{10} + 1857788 x^{9} + 703905 x^{8} - 369684 x^{7} - 102396 x^{6} + 43393 x^{5} + 8232 x^{4} - 2512 x^{3} - 288 x^{2} + 48 x + 1 \)

Invariants

Degree:		$40$
Signature:		$[40, 0]$
Discriminant:		\(100383397447978918530459891214693626269465146363712847232818603515625=3^{20}\cdot 5^{30}\cdot 11^{36}\)
Root discriminant:		$50.12$
Ramified primes:		$3, 5, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(165=3\cdot 5\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{165}(1,·)$, $\chi_{165}(131,·)$, $\chi_{165}(4,·)$, $\chi_{165}(134,·)$, $\chi_{165}(7,·)$, $\chi_{165}(136,·)$, $\chi_{165}(137,·)$, $\chi_{165}(13,·)$, $\chi_{165}(142,·)$, $\chi_{165}(16,·)$, $\chi_{165}(149,·)$, $\chi_{165}(23,·)$, $\chi_{165}(152,·)$, $\chi_{165}(28,·)$, $\chi_{165}(29,·)$, $\chi_{165}(158,·)$, $\chi_{165}(31,·)$, $\chi_{165}(161,·)$, $\chi_{165}(34,·)$, $\chi_{165}(164,·)$, $\chi_{165}(38,·)$, $\chi_{165}(41,·)$, $\chi_{165}(43,·)$, $\chi_{165}(47,·)$, $\chi_{165}(49,·)$, $\chi_{165}(52,·)$, $\chi_{165}(53,·)$, $\chi_{165}(64,·)$, $\chi_{165}(73,·)$, $\chi_{165}(74,·)$, $\chi_{165}(91,·)$, $\chi_{165}(92,·)$, $\chi_{165}(101,·)$, $\chi_{165}(112,·)$, $\chi_{165}(113,·)$, $\chi_{165}(116,·)$, $\chi_{165}(118,·)$, $\chi_{165}(122,·)$, $\chi_{165}(124,·)$, $\chi_{165}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$39$
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:		\( 1967499665744724500000 \) (assuming GRH)

Galois group

$C_2\times C_{20}$ (as 40T2):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2\times C_{20}$

Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{5}, \sqrt{33})\), 4.4.15125.1, \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{11})^+\), 8.8.18530015625.1, \(\Q(\zeta_{33})^+\), 10.10.669871503125.1, 10.10.1790566527853125.1, 20.20.3206128490667995866421572265625.1, \(\Q(\zeta_{55})^+\), 20.20.82802905234194108120391845703125.1

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	$20^{2}$	R	R	$20^{2}$	R	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/29.5.0.1}{5} }^{8}$	${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$	$20^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3	Data not computed
5	Data not computed
11	Data not computed