Global number field 44.0.48891877682180103607391812819535352418736437208892474989920547427561.1: \(\Q(\zeta_{69})\)

• Label: 44.0.48891877682...7561.1
• Degree: $44$
• Signature: $[0, 22]$
• Discriminant: $3^{22}\cdot 23^{42}$
• Root discriminant: $34.55$
• Ramified primes: $3, 23$
• Class number: $69$ (GRH)
• Class group: $[69]$ (GRH)
• Galois group: $C_2\times C_{22}$ (as 44T2)

Normalized defining polynomial

\( x^{44} - x^{43} + x^{41} - x^{40} + x^{38} - x^{37} + x^{35} - x^{34} + x^{32} - x^{31} + x^{29} - x^{28} + x^{26} - x^{25} + x^{23} - x^{22} + x^{21} - x^{19} + x^{18} - x^{16} + x^{15} - x^{13} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)

Invariants

Degree:		$44$
Signature:		$[0, 22]$
Discriminant:		\(48891877682180103607391812819535352418736437208892474989920547427561=3^{22}\cdot 23^{42}\)
Root discriminant:		$34.55$
Ramified primes:		$3, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(69=3\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{69}(1,·)$, $\chi_{69}(2,·)$, $\chi_{69}(4,·)$, $\chi_{69}(5,·)$, $\chi_{69}(7,·)$, $\chi_{69}(8,·)$, $\chi_{69}(10,·)$, $\chi_{69}(11,·)$, $\chi_{69}(13,·)$, $\chi_{69}(14,·)$, $\chi_{69}(16,·)$, $\chi_{69}(17,·)$, $\chi_{69}(19,·)$, $\chi_{69}(20,·)$, $\chi_{69}(22,·)$, $\chi_{69}(25,·)$, $\chi_{69}(26,·)$, $\chi_{69}(28,·)$, $\chi_{69}(29,·)$, $\chi_{69}(31,·)$, $\chi_{69}(32,·)$, $\chi_{69}(34,·)$, $\chi_{69}(35,·)$, $\chi_{69}(37,·)$, $\chi_{69}(38,·)$, $\chi_{69}(40,·)$, $\chi_{69}(41,·)$, $\chi_{69}(43,·)$, $\chi_{69}(44,·)$, $\chi_{69}(47,·)$, $\chi_{69}(49,·)$, $\chi_{69}(50,·)$, $\chi_{69}(52,·)$, $\chi_{69}(53,·)$, $\chi_{69}(55,·)$, $\chi_{69}(56,·)$, $\chi_{69}(58,·)$, $\chi_{69}(59,·)$, $\chi_{69}(61,·)$, $\chi_{69}(62,·)$, $\chi_{69}(64,·)$, $\chi_{69}(65,·)$, $\chi_{69}(67,·)$, $\chi_{69}(68,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$

Class group and class number

$C_{69}$, which has order $69$ (assuming GRH)

Unit group

Rank:		$21$
Torsion generator:		\( -a \) (order $138$)
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:		\( 5770465240890302.0 \) (assuming GRH)

Galois group

$C_2\times C_{22}$ (as 44T2):

An abelian group of order 44

The 44 conjugacy class representatives for $C_2\times C_{22}$

Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{69}) \), \(\Q(\sqrt{-3}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.0.304011857053427966889939263171547.1, \(\Q(\zeta_{23})\), \(\Q(\zeta_{69})^+\)

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	$22^{2}$	R	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	R	$22^{2}$	${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$	$22^{2}$	$22^{2}$

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
3	Data not computed
23	Data not computed