Global number field 40.0.32473210254684090614318847656250000000000000000000000000000000000000000.3

• Label: 40.0.32473210254...0000.3
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $2^{40}\cdot 3^{20}\cdot 5^{70}$
• Root discriminant: $57.91$
• Ramified primes: $2, 3, 5$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} + 243 x^{30} + 59049 x^{20} + 14348907 x^{10} + 3486784401 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(32473210254684090614318847656250000000000000000000000000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{70}\)
Root discriminant:		$57.91$
Ramified primes:		$2, 3, 5$
This field is Galois and abelian over $\Q$.
Conductor:		\(300=2^{2}\cdot 3\cdot 5^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{300}(1,·)$, $\chi_{300}(131,·)$, $\chi_{300}(133,·)$, $\chi_{300}(263,·)$, $\chi_{300}(11,·)$, $\chi_{300}(13,·)$, $\chi_{300}(143,·)$, $\chi_{300}(277,·)$, $\chi_{300}(23,·)$, $\chi_{300}(157,·)$, $\chi_{300}(287,·)$, $\chi_{300}(289,·)$, $\chi_{300}(37,·)$, $\chi_{300}(167,·)$, $\chi_{300}(169,·)$, $\chi_{300}(299,·)$, $\chi_{300}(47,·)$, $\chi_{300}(49,·)$, $\chi_{300}(179,·)$, $\chi_{300}(181,·)$, $\chi_{300}(59,·)$, $\chi_{300}(61,·)$, $\chi_{300}(191,·)$, $\chi_{300}(193,·)$, $\chi_{300}(71,·)$, $\chi_{300}(73,·)$, $\chi_{300}(203,·)$, $\chi_{300}(83,·)$, $\chi_{300}(217,·)$, $\chi_{300}(97,·)$, $\chi_{300}(227,·)$, $\chi_{300}(229,·)$, $\chi_{300}(107,·)$, $\chi_{300}(109,·)$, $\chi_{300}(239,·)$, $\chi_{300}(241,·)$, $\chi_{300}(119,·)$, $\chi_{300}(121,·)$, $\chi_{300}(251,·)$, $\chi_{300}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$, $\frac{1}{43046721} a^{32}$, $\frac{1}{43046721} a^{33}$, $\frac{1}{129140163} a^{34}$, $\frac{1}{129140163} a^{35}$, $\frac{1}{387420489} a^{36}$, $\frac{1}{387420489} a^{37}$, $\frac{1}{1162261467} a^{38}$, $\frac{1}{1162261467} a^{39}$

Class group and class number

Not computed

Unit group

Rank:		$19$
Torsion generator:		\( \frac{1}{129140163} a^{34} + \frac{1}{531441} a^{24} + \frac{1}{2187} a^{14} + \frac{1}{9} a^{4} \) (order $50$)
Fundamental units:		Not computed
Regulator:		Not computed

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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\zeta_{5})\), 4.0.18000.1, 5.5.390625.1, 8.0.324000000.3, 10.10.37968750000000000.1, \(\Q(\zeta_{25})^+\), 10.10.189843750000000000.1, 20.20.36040649414062500000000000000000000.1, \(\Q(\zeta_{25})\), 20.0.180203247070312500000000000000000000.1

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	R	R	${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/11.5.0.1}{5} }^{8}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
2	Data not computed
3	Data not computed
5	Data not computed