Number field 44.0.77585475977450084659036490313653067375988247701832988519181199114271465890241714493487961590290069580078125.1

• Label: 44.0.775...125.1
• Degree: $44$
• Signature: $[0, 22]$
• Discriminant: $7.759\times 10^{106}$
• Root discriminant: \(268.73\)
• Ramified primes: $5,89$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{44}$ (as 44T1)

Normalized defining polynomial

x^44 - x^43 + 46*x^42 - 47*x^41 + 861*x^40 - 2333*x^39 + 12302*x^38 - 63990*x^37 + 133754*x^36 - 1021468*x^35 + 1930750*x^34 - 10014031*x^33 + 34478590*x^32 - 55120652*x^31 + 488448869*x^30 - 4801625*x^29 + 3550403737*x^28 - 3384252805*x^27 + 5825255746*x^26 - 28254892927*x^25 + 51190951460*x^24 - 144055141516*x^23 + 209880892930*x^22 - 3322585528251*x^21 + 9644185539226*x^20 - 31781584647001*x^19 + 93774885990329*x^18 - 268079177298866*x^17 + 734100939620087*x^16 - 1518218568490401*x^15 + 3168243981907192*x^14 - 5634610133492386*x^13 + 11148716367452975*x^12 - 19690479891452526*x^11 + 33905835107525892*x^10 - 49091119860089206*x^9 + 64926814287050439*x^8 - 72968813297711373*x^7 + 76402710809874371*x^6 - 68958090177933733*x^5 + 57707011951741390*x^4 - 39468700453637824*x^3 + 23684548337191503*x^2 - 9522126521026368*x + 3033550844805431

Invariants

Degree:		$44$
Signature:		$[0, 22]$
Discriminant:		77585475977450084659036490313653067375988247701832988519181199114271465890241714493487961590290069580078125
Root discriminant:		\(268.73\)
Galois root discriminant:		$5^{3/4}89^{43/44}\approx 268.72816808491586$
Ramified primes:		\(5\), \(89\)
Discriminant root field:		\(\Q(\sqrt{445}) \)
$\card{ \Gal(K/\Q) }$:		$44$
This field is Galois and abelian over $\Q$.
Conductor:		\(445=5\cdot 89\)
Dirichlet character group:		$\lbrace$$\chi_{445}(256,·)$, $\chi_{445}(1,·)$, $\chi_{445}(258,·)$, $\chi_{445}(259,·)$, $\chi_{445}(262,·)$, $\chi_{445}(257,·)$, $\chi_{445}(392,·)$, $\chi_{445}(139,·)$, $\chi_{445}(271,·)$, $\chi_{445}(16,·)$, $\chi_{445}(401,·)$, $\chi_{445}(403,·)$, $\chi_{445}(68,·)$, $\chi_{445}(156,·)$, $\chi_{445}(289,·)$, $\chi_{445}(42,·)$, $\chi_{445}(44,·)$, $\chi_{445}(429,·)$, $\chi_{445}(174,·)$, $\chi_{445}(306,·)$, $\chi_{445}(53,·)$, $\chi_{445}(183,·)$, $\chi_{445}(186,·)$, $\chi_{445}(187,·)$, $\chi_{445}(188,·)$, $\chi_{445}(189,·)$, $\chi_{445}(322,·)$, $\chi_{445}(324,·)$, $\chi_{445}(198,·)$, $\chi_{445}(72,·)$, $\chi_{445}(331,·)$, $\chi_{445}(338,·)$, $\chi_{445}(121,·)$, $\chi_{445}(218,·)$, $\chi_{445}(91,·)$, $\chi_{445}(354,·)$, $\chi_{445}(227,·)$, $\chi_{445}(444,·)$, $\chi_{445}(107,·)$, $\chi_{445}(114,·)$, $\chi_{445}(373,·)$, $\chi_{445}(247,·)$, $\chi_{445}(377,·)$, $\chi_{445}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{2097152}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $\frac{1}{16909}a^{42}+\frac{6424}{16909}a^{41}+\frac{6322}{16909}a^{40}-\frac{4786}{16909}a^{39}+\frac{1899}{16909}a^{38}-\frac{5232}{16909}a^{37}-\frac{5580}{16909}a^{36}+\frac{427}{16909}a^{35}+\frac{11}{37}a^{34}+\frac{1797}{16909}a^{33}-\frac{4568}{16909}a^{32}+\frac{2044}{16909}a^{31}-\frac{7930}{16909}a^{30}+\frac{8184}{16909}a^{29}-\frac{7441}{16909}a^{28}+\frac{580}{16909}a^{27}+\frac{6777}{16909}a^{26}+\frac{4308}{16909}a^{25}-\frac{5732}{16909}a^{24}+\frac{2087}{16909}a^{23}-\frac{2498}{16909}a^{22}+\frac{2001}{16909}a^{21}-\frac{1063}{16909}a^{20}-\frac{1593}{16909}a^{19}+\frac{5219}{16909}a^{18}+\frac{5611}{16909}a^{17}-\frac{1888}{16909}a^{16}+\frac{6534}{16909}a^{15}+\frac{3315}{16909}a^{14}+\frac{6156}{16909}a^{13}-\frac{668}{16909}a^{12}+\frac{3421}{16909}a^{11}-\frac{4338}{16909}a^{10}+\frac{8010}{16909}a^{9}-\frac{8308}{16909}a^{8}-\frac{5570}{16909}a^{7}-\frac{1269}{16909}a^{6}-\frac{3981}{16909}a^{5}-\frac{4269}{16909}a^{4}-\frac{5051}{16909}a^{3}-\frac{997}{16909}a^{2}+\frac{96}{16909}a+\frac{276}{16909}$, $\frac{1}{56\!\cdots\!21}a^{43}-\frac{71\!\cdots\!95}{56\!\cdots\!21}a^{42}+\frac{13\!\cdots\!26}{56\!\cdots\!21}a^{41}+\frac{16\!\cdots\!42}{56\!\cdots\!21}a^{40}+\frac{23\!\cdots\!64}{56\!\cdots\!21}a^{39}-\frac{13\!\cdots\!60}{56\!\cdots\!21}a^{38}-\frac{65\!\cdots\!12}{15\!\cdots\!33}a^{37}-\frac{25\!\cdots\!18}{56\!\cdots\!21}a^{36}+\frac{17\!\cdots\!45}{56\!\cdots\!21}a^{35}-\frac{10\!\cdots\!23}{56\!\cdots\!21}a^{34}+\frac{16\!\cdots\!98}{56\!\cdots\!21}a^{33}+\frac{43\!\cdots\!49}{56\!\cdots\!21}a^{32}-\frac{21\!\cdots\!99}{56\!\cdots\!21}a^{31}+\frac{22\!\cdots\!47}{56\!\cdots\!21}a^{30}-\frac{51\!\cdots\!52}{56\!\cdots\!21}a^{29}-\frac{57\!\cdots\!04}{56\!\cdots\!21}a^{28}-\frac{47\!\cdots\!16}{56\!\cdots\!21}a^{27}+\frac{31\!\cdots\!75}{56\!\cdots\!21}a^{26}-\frac{55\!\cdots\!20}{56\!\cdots\!21}a^{25}-\frac{22\!\cdots\!40}{56\!\cdots\!21}a^{24}-\frac{50\!\cdots\!32}{56\!\cdots\!21}a^{23}+\frac{15\!\cdots\!66}{56\!\cdots\!21}a^{22}-\frac{25\!\cdots\!41}{56\!\cdots\!21}a^{21}-\frac{15\!\cdots\!01}{56\!\cdots\!21}a^{20}+\frac{23\!\cdots\!79}{56\!\cdots\!21}a^{19}-\frac{47\!\cdots\!62}{56\!\cdots\!21}a^{18}-\frac{17\!\cdots\!14}{56\!\cdots\!21}a^{17}+\frac{18\!\cdots\!02}{56\!\cdots\!21}a^{16}+\frac{26\!\cdots\!18}{56\!\cdots\!21}a^{15}-\frac{12\!\cdots\!16}{56\!\cdots\!21}a^{14}+\frac{79\!\cdots\!68}{56\!\cdots\!21}a^{13}-\frac{22\!\cdots\!62}{56\!\cdots\!21}a^{12}-\frac{85\!\cdots\!28}{56\!\cdots\!21}a^{11}-\frac{49\!\cdots\!75}{56\!\cdots\!21}a^{10}+\frac{16\!\cdots\!05}{56\!\cdots\!21}a^{9}+\frac{40\!\cdots\!01}{56\!\cdots\!21}a^{8}-\frac{17\!\cdots\!58}{56\!\cdots\!21}a^{7}-\frac{29\!\cdots\!39}{15\!\cdots\!33}a^{6}-\frac{95\!\cdots\!34}{56\!\cdots\!21}a^{5}+\frac{12\!\cdots\!58}{56\!\cdots\!21}a^{4}-\frac{36\!\cdots\!83}{56\!\cdots\!21}a^{3}+\frac{77\!\cdots\!29}{56\!\cdots\!21}a^{2}-\frac{19\!\cdots\!25}{56\!\cdots\!21}a-\frac{82\!\cdots\!47}{44\!\cdots\!87}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$21$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

Galois group

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

A cyclic group of order 44

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

Character table for $C_{44}$

Intermediate fields

\(\Q(\sqrt{445}) \), 4.0.88121125.2, 11.11.31181719929966183601.1, 22.22.4225325661298128736141932434930999908246533203125.1

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	$44$	$22^{2}$	R	${\href{/padicField/7.11.0.1}{11} }^{4}$	${\href{/padicField/11.11.0.1}{11} }^{4}$	${\href{/padicField/13.11.0.1}{11} }^{4}$	$44$	$44$	$22^{2}$	$44$	$44$	${\href{/padicField/37.2.0.1}{2} }^{22}$	$44$	$22^{2}$	$44$	$44$	$44$

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
\(5\)		Deg $44$	$4$	$11$	$33$
\(89\)		Deg $44$	$44$	$1$	$43$