Global number field 40.0.2806502314667513790456400791754773417894971971642777758942872517738496.1

• Label: 40.0.28065023146...8496.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $2^{120}\cdot 11^{32}$
• Root discriminant: $54.48$
• Ramified primes: $2, 11$
• Class number: $2605$ (GRH)
• Class group: $[2605]$ (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} + 257 x^{32} + 14016 x^{24} + 105419 x^{16} + 23219 x^{8} + 1 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(2806502314667513790456400791754773417894971971642777758942872517738496=2^{120}\cdot 11^{32}\)
Root discriminant:		$54.48$
Ramified primes:		$2, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(176=2^{4}\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(3,·)$, $\chi_{176}(133,·)$, $\chi_{176}(135,·)$, $\chi_{176}(9,·)$, $\chi_{176}(141,·)$, $\chi_{176}(15,·)$, $\chi_{176}(147,·)$, $\chi_{176}(23,·)$, $\chi_{176}(25,·)$, $\chi_{176}(155,·)$, $\chi_{176}(157,·)$, $\chi_{176}(5,·)$, $\chi_{176}(27,·)$, $\chi_{176}(37,·)$, $\chi_{176}(31,·)$, $\chi_{176}(169,·)$, $\chi_{176}(45,·)$, $\chi_{176}(47,·)$, $\chi_{176}(49,·)$, $\chi_{176}(53,·)$, $\chi_{176}(137,·)$, $\chi_{176}(159,·)$, $\chi_{176}(67,·)$, $\chi_{176}(69,·)$, $\chi_{176}(71,·)$, $\chi_{176}(75,·)$, $\chi_{176}(81,·)$, $\chi_{176}(163,·)$, $\chi_{176}(89,·)$, $\chi_{176}(91,·)$, $\chi_{176}(93,·)$, $\chi_{176}(97,·)$, $\chi_{176}(59,·)$, $\chi_{176}(103,·)$, $\chi_{176}(111,·)$, $\chi_{176}(113,·)$, $\chi_{176}(115,·)$, $\chi_{176}(119,·)$, $\chi_{176}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{22148330043533} a^{32} + \frac{2726870929642}{22148330043533} a^{24} - \frac{7580340779727}{22148330043533} a^{16} + \frac{6188800161857}{22148330043533} a^{8} + \frac{6246971138049}{22148330043533}$, $\frac{1}{22148330043533} a^{33} + \frac{2726870929642}{22148330043533} a^{25} - \frac{7580340779727}{22148330043533} a^{17} + \frac{6188800161857}{22148330043533} a^{9} + \frac{6246971138049}{22148330043533} a$, $\frac{1}{22148330043533} a^{34} + \frac{2726870929642}{22148330043533} a^{26} - \frac{7580340779727}{22148330043533} a^{18} + \frac{6188800161857}{22148330043533} a^{10} + \frac{6246971138049}{22148330043533} a^{2}$, $\frac{1}{22148330043533} a^{35} + \frac{2726870929642}{22148330043533} a^{27} - \frac{7580340779727}{22148330043533} a^{19} + \frac{6188800161857}{22148330043533} a^{11} + \frac{6246971138049}{22148330043533} a^{3}$, $\frac{1}{22148330043533} a^{36} + \frac{2726870929642}{22148330043533} a^{28} - \frac{7580340779727}{22148330043533} a^{20} + \frac{6188800161857}{22148330043533} a^{12} + \frac{6246971138049}{22148330043533} a^{4}$, $\frac{1}{22148330043533} a^{37} + \frac{2726870929642}{22148330043533} a^{29} - \frac{7580340779727}{22148330043533} a^{21} + \frac{6188800161857}{22148330043533} a^{13} + \frac{6246971138049}{22148330043533} a^{5}$, $\frac{1}{22148330043533} a^{38} + \frac{2726870929642}{22148330043533} a^{30} - \frac{7580340779727}{22148330043533} a^{22} + \frac{6188800161857}{22148330043533} a^{14} + \frac{6246971138049}{22148330043533} a^{6}$, $\frac{1}{22148330043533} a^{39} + \frac{2726870929642}{22148330043533} a^{31} - \frac{7580340779727}{22148330043533} a^{23} + \frac{6188800161857}{22148330043533} a^{15} + \frac{6246971138049}{22148330043533} a^{7}$

Class group and class number

$C_{2605}$, which has order $2605$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( \frac{43114401572}{22148330043533} a^{35} + \frac{11079877784514}{22148330043533} a^{27} + \frac{604158913402484}{22148330043533} a^{19} + \frac{4538168045437461}{22148330043533} a^{11} + \frac{960702298577351}{22148330043533} a^{3} \) (order $16$)
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:		\( 4506035972876552.0 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 4.0.2048.2, \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{16})\), 10.0.219503494144.1, 10.10.7024111812608.1, 10.0.7024111812608.1, 20.0.50522262278163705147147943936.1, 20.0.1655513490330868290261743826894848.1, 20.20.1655513490330868290261743826894848.1

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	$20^{2}$	${\href{/LocalNumberField/17.5.0.1}{5} }^{8}$	$20^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$	$20^{2}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$	$20^{2}$	$20^{2}$

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