Global number field 40.0.10240000000000000000000000000000000000000000000000000000000000000000000000.1

• Label: 40.0.10240000000...0000.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $2^{80}\cdot 5^{70}$
• Root discriminant: $66.87$
• Ramified primes: $2, 5$
• Class number: $542102$ (GRH)
• Class group: $[542102]$ (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} + 40 x^{38} + 740 x^{36} + 8400 x^{34} + 65450 x^{32} + 371008 x^{30} + 1582240 x^{28} + 5178240 x^{26} + 13147875 x^{24} + 26013000 x^{22} + 40060019 x^{20} + 47720380 x^{18} + 43459480 x^{16} + 29715200 x^{14} + 14855725 x^{12} + 5226012 x^{10} + 1221495 x^{8} + 172920 x^{6} + 12475 x^{4} + 300 x^{2} + 1 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(10240000000000000000000000000000000000000000000000000000000000000000000000=2^{80}\cdot 5^{70}\)
Root discriminant:		$66.87$
Ramified primes:		$2, 5$
This field is Galois and abelian over $\Q$.
Conductor:		\(200=2^{3}\cdot 5^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(131,·)$, $\chi_{200}(133,·)$, $\chi_{200}(7,·)$, $\chi_{200}(9,·)$, $\chi_{200}(11,·)$, $\chi_{200}(13,·)$, $\chi_{200}(143,·)$, $\chi_{200}(19,·)$, $\chi_{200}(23,·)$, $\chi_{200}(179,·)$, $\chi_{200}(157,·)$, $\chi_{200}(161,·)$, $\chi_{200}(37,·)$, $\chi_{200}(167,·)$, $\chi_{200}(129,·)$, $\chi_{200}(41,·)$, $\chi_{200}(171,·)$, $\chi_{200}(173,·)$, $\chi_{200}(47,·)$, $\chi_{200}(49,·)$, $\chi_{200}(51,·)$, $\chi_{200}(53,·)$, $\chi_{200}(183,·)$, $\chi_{200}(59,·)$, $\chi_{200}(63,·)$, $\chi_{200}(139,·)$, $\chi_{200}(197,·)$, $\chi_{200}(77,·)$, $\chi_{200}(81,·)$, $\chi_{200}(87,·)$, $\chi_{200}(89,·)$, $\chi_{200}(91,·)$, $\chi_{200}(93,·)$, $\chi_{200}(99,·)$, $\chi_{200}(103,·)$, $\chi_{200}(117,·)$, $\chi_{200}(169,·)$, $\chi_{200}(121,·)$, $\chi_{200}(127,·)$$\rbrace$
This is a CM field.

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

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

Class group and class number

$C_{542102}$, which has order $542102$ (assuming GRH)

Unit group

Rank:		$19$
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:		\( 144312257071955.8 \) (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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.0.8000.2, \(\Q(\zeta_{20})^+\), 5.5.390625.1, 8.0.1024000000.1, 10.0.5000000000000000.1, \(\Q(\zeta_{25})^+\), 10.0.25000000000000000.1, 20.0.625000000000000000000000000000000.1, 20.0.3125000000000000000000000000000000.1, \(\Q(\zeta_{100})^+\)

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	$20^{2}$	R	${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/19.5.0.1}{5} }^{8}$	$20^{2}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/59.5.0.1}{5} }^{8}$

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
5	Data not computed