Global number field 44.44.860115008245742907292219227824365111518443501386754869093198564719785672888549376.1: \(\Q(\zeta_{276})^+\)

• Label: 44.44.8601150082...9376.1
• Degree: $44$
• Signature: $[44, 0]$
• Discriminant: $2^{44}\cdot 3^{22}\cdot 23^{42}$
• Root discriminant: $69.09$
• Ramified primes: $2, 3, 23$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2\times C_{22}$ (as 44T2)

Normalized defining polynomial

\( x^{44} - 43 x^{42} + 860 x^{40} - 10622 x^{38} + 90724 x^{36} - 568616 x^{34} + 2708289 x^{32} - 10016751 x^{30} + 29148711 x^{28} - 67217827 x^{26} + 123140835 x^{24} - 178931535 x^{22} + 205096155 x^{20} - 183670815 x^{18} + 126676230 x^{16} - 65934180 x^{14} + 25178895 x^{12} - 6781005 x^{10} + 1216380 x^{8} - 133210 x^{6} + 7700 x^{4} - 176 x^{2} + 1 \)

Invariants

Degree:		$44$
Signature:		$[44, 0]$
Discriminant:		\(860115008245742907292219227824365111518443501386754869093198564719785672888549376=2^{44}\cdot 3^{22}\cdot 23^{42}\)
Root discriminant:		$69.09$
Ramified primes:		$2, 3, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(276=2^{2}\cdot 3\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{276}(1,·)$, $\chi_{276}(131,·)$, $\chi_{276}(5,·)$, $\chi_{276}(7,·)$, $\chi_{276}(265,·)$, $\chi_{276}(13,·)$, $\chi_{276}(17,·)$, $\chi_{276}(19,·)$, $\chi_{276}(149,·)$, $\chi_{276}(25,·)$, $\chi_{276}(47,·)$, $\chi_{276}(133,·)$, $\chi_{276}(35,·)$, $\chi_{276}(65,·)$, $\chi_{276}(167,·)$, $\chi_{276}(169,·)$, $\chi_{276}(43,·)$, $\chi_{276}(175,·)$, $\chi_{276}(49,·)$, $\chi_{276}(179,·)$, $\chi_{276}(53,·)$, $\chi_{276}(137,·)$, $\chi_{276}(59,·)$, $\chi_{276}(193,·)$, $\chi_{276}(67,·)$, $\chi_{276}(71,·)$, $\chi_{276}(73,·)$, $\chi_{276}(119,·)$, $\chi_{276}(79,·)$, $\chi_{276}(85,·)$, $\chi_{276}(215,·)$, $\chi_{276}(89,·)$, $\chi_{276}(91,·)$, $\chi_{276}(221,·)$, $\chi_{276}(199,·)$, $\chi_{276}(95,·)$, $\chi_{276}(103,·)$, $\chi_{276}(235,·)$, $\chi_{276}(239,·)$, $\chi_{276}(113,·)$, $\chi_{276}(245,·)$, $\chi_{276}(247,·)$, $\chi_{276}(121,·)$, $\chi_{276}(125,·)$$\rbrace$
This is not 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

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$43$
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:		\( 396826934994876900000000000 \) (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

\(\Q(\sqrt{69}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{3}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), \(\Q(\zeta_{69})^+\), \(\Q(\zeta_{92})^+\), 22.22.1275118148086621135238339811277472268288.1

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	$22^{2}$	$22^{2}$	${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$	${\href{/LocalNumberField/13.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	R	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$	$22^{2}$	$22^{2}$

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
2	Data not computed
3	Data not computed
23	Data not computed