Global number field 40.0.37371137649685869661036271274571515833850102932794388078657536.1: \(\Q(\zeta_{88})\)

• Label: 40.0.37371137649...7536.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $2^{80}\cdot 11^{36}$
• Root discriminant: $34.62$
• Ramified primes: $2, 11$
• Class number: $55$ (GRH)
• Class group: $[55]$ (GRH)
• Galois group: $C_2^2\times C_{10}$ (as 40T7)

Normalized defining polynomial

\( x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(37371137649685869661036271274571515833850102932794388078657536=2^{80}\cdot 11^{36}\)
Root discriminant:		$34.62$
Ramified primes:		$2, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(88=2^{3}\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{88}(1,·)$, $\chi_{88}(3,·)$, $\chi_{88}(5,·)$, $\chi_{88}(7,·)$, $\chi_{88}(9,·)$, $\chi_{88}(13,·)$, $\chi_{88}(15,·)$, $\chi_{88}(17,·)$, $\chi_{88}(19,·)$, $\chi_{88}(21,·)$, $\chi_{88}(23,·)$, $\chi_{88}(25,·)$, $\chi_{88}(27,·)$, $\chi_{88}(29,·)$, $\chi_{88}(31,·)$, $\chi_{88}(35,·)$, $\chi_{88}(37,·)$, $\chi_{88}(39,·)$, $\chi_{88}(41,·)$, $\chi_{88}(43,·)$, $\chi_{88}(45,·)$, $\chi_{88}(47,·)$, $\chi_{88}(49,·)$, $\chi_{88}(51,·)$, $\chi_{88}(53,·)$, $\chi_{88}(57,·)$, $\chi_{88}(59,·)$, $\chi_{88}(61,·)$, $\chi_{88}(63,·)$, $\chi_{88}(65,·)$, $\chi_{88}(67,·)$, $\chi_{88}(69,·)$, $\chi_{88}(71,·)$, $\chi_{88}(73,·)$, $\chi_{88}(75,·)$, $\chi_{88}(79,·)$, $\chi_{88}(81,·)$, $\chi_{88}(83,·)$, $\chi_{88}(85,·)$, $\chi_{88}(87,·)$$\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

$C_{55}$, which has order $55$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( a \) (order $88$)
Fundamental units:		\( a^{8} + 1 \),  \( a^{16} + 1 \),  \( a^{16} + a^{8} + 1 \),  \( a^{8} - a^{4} \),  \( a^{6} - 1 \),  \( a^{18} - 1 \),  \( a^{10} - 1 \),  \( a^{14} - 1 \),  \( a^{3} - 1 \),  \( a^{15} - 1 \),  \( a^{5} - 1 \),  \( a^{9} - 1 \),  \( a^{17} - 1 \),  \( a^{7} - 1 \),  \( a - 1 \),  \( a^{7} + a^{6} \),  \( a^{13} - 1 \),  \( a^{22} + a^{11} + 1 \),  \( a^{19} - 1 \) (assuming GRH)
Regulator:		\( 172972974175441.34 \) (assuming GRH)

Galois group

$C_2^2\times C_{10}$ (as 40T7):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2^2\times C_{10}$

Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{22}) \), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{22})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.0.959512576.1, 10.10.7024111812608.1, 10.0.219503494144.1, 10.0.7024111812608.1, \(\Q(\zeta_{11})\), 10.0.77265229938688.1, \(\Q(\zeta_{44})^+\), 10.10.77265229938688.1, 20.0.50522262278163705147147943936.1, 20.0.5969915757478328440239161344.6, \(\Q(\zeta_{88})^+\), \(\Q(\zeta_{44})\), 20.0.6113193735657808322804901216256.4, 20.0.5969915757478328440239161344.5, 20.0.6113193735657808322804901216256.3

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	${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$	R	${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
11	Data not computed