Number field 44.44.181375764426442332776050749828434842919201892356144879261148162186145782470703125.1: \(\Q(\zeta_{115})^+\)

• Label: 44.44.181...125.1
• Degree: $44$
• Signature: $[44, 0]$
• Discriminant: $1.814\times 10^{80}$
• Root discriminant: $66.69$
• Ramified primes: $5, 23$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{44}$ (as 44T1)

Normalized defining polynomial

\(x^{44} - x^{43} - 44 x^{42} + 43 x^{41} + 902 x^{40} - 859 x^{39} - 11441 x^{38} + 10582 x^{37} + 100567 x^{36} - 89985 x^{35} - 650236 x^{34} + 560251 x^{33} + 3203650 x^{32} - 2643399 x^{31} - 12294569 x^{30} + 9651170 x^{29} + 37253225 x^{28} - 27602055 x^{27} - 89808680 x^{26} + 62206625 x^{25} + 172779011 x^{24} - 110572386 x^{23} - 265002643 x^{22} + 154430258 x^{21} + 322457859 x^{20} - 168027624 x^{19} - 308473797 x^{18} + 140446403 x^{17} + 228768475 x^{16} - 88323383 x^{15} - 128871012 x^{14} + 40552321 x^{13} + 53558102 x^{12} - 13016729 x^{11} - 15742859 x^{10} + 2742874 x^{9} + 3073662 x^{8} - 347233 x^{7} - 361455 x^{6} + 24089 x^{5} + 21786 x^{4} - 986 x^{3} - 504 x^{2} + 24 x + 1\)  

Invariants

Degree:		$44$
Signature:		$[44, 0]$
Discriminant:		\(181\!\cdots\!125\)\(\medspace = 5^{33}\cdot 23^{42}\)
Root discriminant:		$66.69$
Ramified primes:		$5, 23$
$|\Gal(K/\Q)|$:		$44$
This field is Galois and abelian over $\Q$.
Conductor:		\(115=5\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(7,·)$, $\chi_{115}(9,·)$, $\chi_{115}(16,·)$, $\chi_{115}(17,·)$, $\chi_{115}(22,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(28,·)$, $\chi_{115}(29,·)$, $\chi_{115}(31,·)$, $\chi_{115}(33,·)$, $\chi_{115}(36,·)$, $\chi_{115}(37,·)$, $\chi_{115}(38,·)$, $\chi_{115}(39,·)$, $\chi_{115}(41,·)$, $\chi_{115}(42,·)$, $\chi_{115}(43,·)$, $\chi_{115}(49,·)$, $\chi_{115}(53,·)$, $\chi_{115}(54,·)$, $\chi_{115}(57,·)$, $\chi_{115}(59,·)$, $\chi_{115}(63,·)$, $\chi_{115}(64,·)$, $\chi_{115}(67,·)$, $\chi_{115}(68,·)$, $\chi_{115}(71,·)$, $\chi_{115}(81,·)$, $\chi_{115}(83,·)$, $\chi_{115}(88,·)$, $\chi_{115}(94,·)$, $\chi_{115}(96,·)$, $\chi_{115}(97,·)$, $\chi_{115}(101,·)$, $\chi_{115}(102,·)$, $\chi_{115}(103,·)$, $\chi_{115}(104,·)$, $\chi_{115}(107,·)$, $\chi_{115}(112,·)$, $\chi_{115}(113,·)$$\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:		\( 183282479706502550000000000 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 183282479706502550000000000 \cdot 1}{2\sqrt{181375764426442332776050749828434842919201892356144879261148162186145782470703125}}\approx 0.119707438429760$ (assuming GRH)

Galois group

$C_{44}$ (as 44T1):

A cyclic group of order 44

The 44 conjugacy class representatives for $C_{44}$

Character table for $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.66125.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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	$44$	$44$	R	$44$	$22^{2}$	$44$	$44$	${\href{/LocalNumberField/19.11.0.1}{11} }^{4}$	R	$22^{2}$	${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$	$44$	${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$	$44$	${\href{/LocalNumberField/47.4.0.1}{4} }^{11}$	$44$	$22^{2}$

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
5	Data not computed
23	Data not computed