Global number field 40.0.41090000389047069406072163992081637611400284636821909168682596532209319936.2

• Label: 40.0.41090000389...9936.2
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $2^{120}\cdot 11^{36}$
• Root discriminant: $69.24$
• Ramified primes: $2, 11$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} + 385 x^{32} + 38720 x^{24} + 842523 x^{16} + 1449459 x^{8} + 14641 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
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$$\chi_{176}(1,·)$, $\chi_{176}(131,·)$, $\chi_{176}(135,·)$, $\chi_{176}(9,·)$, $\chi_{176}(139,·)$, $\chi_{176}(13,·)$, $\chi_{176}(15,·)$, $\chi_{176}(19,·)$, $\chi_{176}(21,·)$, $\chi_{176}(23,·)$, $\chi_{176}(25,·)$, $\chi_{176}(29,·)$, $\chi_{176}(159,·)$, $\chi_{176}(35,·)$, $\chi_{176}(169,·)$, $\chi_{176}(43,·)$, $\chi_{176}(173,·)$, $\chi_{176}(47,·)$, $\chi_{176}(49,·)$, $\chi_{176}(51,·)$, $\chi_{176}(137,·)$, $\chi_{176}(31,·)$, $\chi_{176}(61,·)$, $\chi_{176}(71,·)$, $\chi_{176}(81,·)$, $\chi_{176}(83,·)$, $\chi_{176}(85,·)$, $\chi_{176}(89,·)$, $\chi_{176}(97,·)$, $\chi_{176}(101,·)$, $\chi_{176}(103,·)$, $\chi_{176}(171,·)$, $\chi_{176}(107,·)$, $\chi_{176}(109,·)$, $\chi_{176}(111,·)$, $\chi_{176}(113,·)$, $\chi_{176}(117,·)$, $\chi_{176}(119,·)$, $\chi_{176}(123,·)$, $\chi_{176}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{11} a^{11}$, $\frac{1}{11} a^{12}$, $\frac{1}{11} a^{13}$, $\frac{1}{11} a^{14}$, $\frac{1}{11} a^{15}$, $\frac{1}{11} a^{16}$, $\frac{1}{11} a^{17}$, $\frac{1}{11} a^{18}$, $\frac{1}{11} a^{19}$, $\frac{1}{121} a^{20}$, $\frac{1}{121} a^{21}$, $\frac{1}{121} a^{22}$, $\frac{1}{121} a^{23}$, $\frac{1}{121} a^{24}$, $\frac{1}{121} a^{25}$, $\frac{1}{121} a^{26}$, $\frac{1}{121} a^{27}$, $\frac{1}{121} a^{28}$, $\frac{1}{121} a^{29}$, $\frac{1}{1331} a^{30}$, $\frac{1}{1331} a^{31}$, $\frac{1}{13990456741151} a^{32} - \frac{1889100866}{1271859703741} a^{24} + \frac{2932669862}{115623609431} a^{16} + \frac{3561712835}{10511237221} a^{8} - \frac{4799947149}{10511237221}$, $\frac{1}{13990456741151} a^{33} - \frac{1889100866}{1271859703741} a^{25} + \frac{2932669862}{115623609431} a^{17} + \frac{3561712835}{10511237221} a^{9} - \frac{4799947149}{10511237221} a$, $\frac{1}{13990456741151} a^{34} - \frac{1889100866}{1271859703741} a^{26} + \frac{2932669862}{115623609431} a^{18} - \frac{2866107699}{115623609431} a^{10} - \frac{4799947149}{10511237221} a^{2}$, $\frac{1}{13990456741151} a^{35} - \frac{1889100866}{1271859703741} a^{27} + \frac{2932669862}{115623609431} a^{19} - \frac{2866107699}{115623609431} a^{11} - \frac{4799947149}{10511237221} a^{3}$, $\frac{1}{13990456741151} a^{36} - \frac{1889100866}{1271859703741} a^{28} + \frac{725656819}{1271859703741} a^{20} - \frac{2866107699}{115623609431} a^{12} - \frac{4799947149}{10511237221} a^{4}$, $\frac{1}{13990456741151} a^{37} - \frac{1889100866}{1271859703741} a^{29} + \frac{725656819}{1271859703741} a^{21} - \frac{2866107699}{115623609431} a^{13} - \frac{4799947149}{10511237221} a^{5}$, $\frac{1}{13990456741151} a^{38} + \frac{242364916}{13990456741151} a^{30} + \frac{725656819}{1271859703741} a^{22} - \frac{2866107699}{115623609431} a^{14} - \frac{4799947149}{10511237221} a^{6}$, $\frac{1}{13990456741151} a^{39} + \frac{242364916}{13990456741151} a^{31} + \frac{725656819}{1271859703741} a^{23} - \frac{2866107699}{115623609431} a^{15} - \frac{4799947149}{10511237221} a^{7}$

Class group and class number

Not computed

Unit group

Rank:		$19$
Torsion generator:		\( -\frac{22017846}{13990456741151} a^{34} - \frac{771266696}{1271859703741} a^{26} - \frac{7068655981}{115623609431} a^{18} - \frac{155782250865}{115623609431} a^{10} - \frac{33250829639}{10511237221} a^{2} \) (order $8$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\), 4.4.247808.1, 4.0.247808.2, \(\Q(\zeta_{11})^+\), 8.0.245635219456.2, 10.0.219503494144.1, 10.10.7024111812608.1, 10.0.7024111812608.1, 20.0.50522262278163705147147943936.1, 20.20.200317132330035063121671003054276608.1, 20.0.200317132330035063121671003054276608.1

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	$20^{2}$	${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$	$20^{2}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$	$20^{2}$	$20^{2}$

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