Global number field 40.40.41090000389047069406072163992081637611400284636821909168682596532209319936.1: \(\Q(\zeta_{176})^+\)

• Label: 40.40.4109000038...9936.1
• Degree: $40$
• Signature: $[40, 0]$
• Discriminant: $2^{120}\cdot 11^{36}$
• Root discriminant: $69.24$
• Ramified primes: $2, 11$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} - 40 x^{38} + 740 x^{36} - 8400 x^{34} + 65449 x^{32} - 370976 x^{30} + 1581776 x^{28} - 5174208 x^{26} + 13124476 x^{24} - 25917344 x^{22} + 39776912 x^{20} - 47106624 x^{18} + 42484835 x^{16} - 28593712 x^{14} + 13942072 x^{12} - 4719648 x^{10} + 1042879 x^{8} - 137208 x^{6} + 9260 x^{4} - 240 x^{2} + 1 \)

Invariants

Degree:		$40$
Signature:		$[40, 0]$
Discriminant:		\(41090000389047069406072163992081637611400284636821909168682596532209319936=2^{120}\cdot 11^{36}\)
Root discriminant:		$69.24$
Ramified primes:		$2, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(176=2^{4}\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(131,·)$, $\chi_{176}(5,·)$, $\chi_{176}(7,·)$, $\chi_{176}(9,·)$, $\chi_{176}(139,·)$, $\chi_{176}(141,·)$, $\chi_{176}(43,·)$, $\chi_{176}(19,·)$, $\chi_{176}(151,·)$, $\chi_{176}(25,·)$, $\chi_{176}(157,·)$, $\chi_{176}(133,·)$, $\chi_{176}(35,·)$, $\chi_{176}(37,·)$, $\chi_{176}(167,·)$, $\chi_{176}(169,·)$, $\chi_{176}(171,·)$, $\chi_{176}(45,·)$, $\chi_{176}(175,·)$, $\chi_{176}(49,·)$, $\chi_{176}(51,·)$, $\chi_{176}(53,·)$, $\chi_{176}(137,·)$, $\chi_{176}(63,·)$, $\chi_{176}(69,·)$, $\chi_{176}(79,·)$, $\chi_{176}(81,·)$, $\chi_{176}(83,·)$, $\chi_{176}(87,·)$, $\chi_{176}(89,·)$, $\chi_{176}(93,·)$, $\chi_{176}(95,·)$, $\chi_{176}(97,·)$, $\chi_{176}(107,·)$, $\chi_{176}(113,·)$, $\chi_{176}(123,·)$, $\chi_{176}(125,·)$, $\chi_{176}(39,·)$, $\chi_{176}(127,·)$$\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:		\( 1209836226640861800000000 \) (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

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\zeta_{16})^+\), 4.4.247808.1, \(\Q(\zeta_{11})^+\), 8.8.245635219456.1, 10.10.77265229938688.1, 10.10.7024111812608.1, \(\Q(\zeta_{44})^+\), \(\Q(\zeta_{88})^+\), 20.20.1655513490330868290261743826894848.1, 20.20.200317132330035063121671003054276608.1

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	R	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$	R	$20^{2}$	${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$	$20^{2}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$	$20^{2}$	$20^{2}$

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
11	Data not computed