Number field 44.44.666454163935483494165986073535521413339908119119439689887653437787720225729135378569.1: \(\Q(\zeta_{89})^+\)

• Label: 44.44.666...569.1
• Degree: $44$
• Signature: $[44, 0]$
• Discriminant: $6.665\times 10^{83}$
• Root discriminant: $80.37$
• Ramified prime: $89$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{44}$ (as 44T1)

Normalized defining polynomial

\( x^{44} - x^{43} - 43 x^{42} + 42 x^{41} + 861 x^{40} - 820 x^{39} - 10660 x^{38} + 9880 x^{37} + 91390 x^{36} - 82251 x^{35} - 575757 x^{34} + 501942 x^{33} + 2760681 x^{32} - 2324784 x^{31} - 10295472 x^{30} + 8347680 x^{29} + 30260340 x^{28} - 23535820 x^{27} - 70607460 x^{26} + 52451256 x^{25} + 131128140 x^{24} - 92561040 x^{23} - 193536720 x^{22} + 129024480 x^{21} + 225792840 x^{20} - 141120525 x^{19} - 206253075 x^{18} + 119759850 x^{17} + 145422675 x^{16} - 77558760 x^{15} - 77558760 x^{14} + 37442160 x^{13} + 30421755 x^{12} - 13037895 x^{11} - 8436285 x^{10} + 3124550 x^{9} + 1562275 x^{8} - 480700 x^{7} - 177100 x^{6} + 42504 x^{5} + 10626 x^{4} - 1771 x^{3} - 253 x^{2} + 22 x + 1 \)

Invariants

Degree:		$44$
Signature:		$[44, 0]$
Discriminant:		\(666\!\cdots\!569\)\(\medspace = 89^{43}\)
Root discriminant:		$80.37$
Ramified primes:		$89$
$|\Gal(K/\Q)|$:		$44$
This field is Galois and abelian over $\Q$.
Conductor:		\(89\)
Dirichlet character group:		$\lbrace$$\chi_{89}(1,·)$, $\chi_{89}(2,·)$, $\chi_{89}(4,·)$, $\chi_{89}(5,·)$, $\chi_{89}(8,·)$, $\chi_{89}(9,·)$, $\chi_{89}(10,·)$, $\chi_{89}(11,·)$, $\chi_{89}(16,·)$, $\chi_{89}(17,·)$, $\chi_{89}(18,·)$, $\chi_{89}(20,·)$, $\chi_{89}(21,·)$, $\chi_{89}(22,·)$, $\chi_{89}(25,·)$, $\chi_{89}(32,·)$, $\chi_{89}(34,·)$, $\chi_{89}(36,·)$, $\chi_{89}(39,·)$, $\chi_{89}(40,·)$, $\chi_{89}(42,·)$, $\chi_{89}(44,·)$, $\chi_{89}(45,·)$, $\chi_{89}(47,·)$, $\chi_{89}(49,·)$, $\chi_{89}(50,·)$, $\chi_{89}(53,·)$, $\chi_{89}(55,·)$, $\chi_{89}(57,·)$, $\chi_{89}(64,·)$, $\chi_{89}(67,·)$, $\chi_{89}(68,·)$, $\chi_{89}(69,·)$, $\chi_{89}(71,·)$, $\chi_{89}(72,·)$, $\chi_{89}(73,·)$, $\chi_{89}(78,·)$, $\chi_{89}(79,·)$, $\chi_{89}(80,·)$, $\chi_{89}(81,·)$, $\chi_{89}(84,·)$, $\chi_{89}(85,·)$, $\chi_{89}(87,·)$, $\chi_{89}(88,·)$$\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:		\( 9928065388501928000000000000 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{44}\cdot(2\pi)^{0}\cdot 9928065388501928000000000000 \cdot 1}{2\sqrt{666454163935483494165986073535521413339908119119439689887653437787720225729135378569}}\approx 0.106971798982240$ (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

\(\Q(\sqrt{89}) \), 4.4.704969.1, 11.11.31181719929966183601.1, 22.22.86534669543385676516186776267386878120889.1

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	${\href{/LocalNumberField/2.11.0.1}{11} }^{4}$	$44$	$22^{2}$	$44$	${\href{/LocalNumberField/11.11.0.1}{11} }^{4}$	$44$	$22^{2}$	$44$	$44$	$44$	$44$	${\href{/LocalNumberField/37.4.0.1}{4} }^{11}$	$44$	$44$	$22^{2}$	$22^{2}$	$44$

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
89	Data not computed