Number field 46.0.41918400957309186825512480921766374958487834428424755931803412278009872045477534434640407562255859375.1

• Label: 46.0.419...375.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-4.192\times 10^{100}$
• Root discriminant: \(153.97\)
• Ramified primes: $3,5,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - 21*x^45 + 257*x^44 - 2275*x^43 + 16384*x^42 - 101148*x^41 + 555868*x^40 - 2773061*x^39 + 12764350*x^38 - 54717566*x^37 + 220411975*x^36 - 838622105*x^35 + 3030581020*x^34 - 10433377545*x^33 + 34346625500*x^32 - 108311066940*x^31 + 328058481150*x^30 - 955269123060*x^29 + 2679603110905*x^28 - 7242985668834*x^27 + 18895590411056*x^26 - 47565631805041*x^25 + 115691372227157*x^24 - 271669498342240*x^23 + 616666273593665*x^22 - 1351191092366848*x^21 + 2861502035274188*x^20 - 5844226298666601*x^19 + 11528422276701890*x^18 - 21892211952456407*x^17 + 40102623357336934*x^16 - 70512951104599839*x^15 + 119377822272400223*x^14 - 193124835740771010*x^13 + 300080474989777953*x^12 - 442454098080721470*x^11 + 624685689856470240*x^10 - 827583193065916590*x^9 + 1046376694212400500*x^8 - 1217672042714811795*x^7 + 1349148741219481645*x^6 - 1325965235556309370*x^5 + 1244120066328799465*x^4 - 953228966874939950*x^3 + 714248269821476266*x^2 - 340555683118656924*x + 181351969023278929

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-41918400957309186825512480921766374958487834428424755931803412278009872045477534434640407562255859375
Root discriminant:		\(153.97\)
Galois root discriminant:		$3^{1/2}5^{1/2}47^{22/23}\approx 153.97264864641676$
Ramified primes:		\(3\), \(5\), \(47\)
Discriminant root field:		\(\Q(\sqrt{-15}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(705=3\cdot 5\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{705}(256,·)$, $\chi_{705}(1,·)$, $\chi_{705}(136,·)$, $\chi_{705}(524,·)$, $\chi_{705}(269,·)$, $\chi_{705}(14,·)$, $\chi_{705}(271,·)$, $\chi_{705}(16,·)$, $\chi_{705}(659,·)$, $\chi_{705}(404,·)$, $\chi_{705}(149,·)$, $\chi_{705}(89,·)$, $\chi_{705}(284,·)$, $\chi_{705}(541,·)$, $\chi_{705}(286,·)$, $\chi_{705}(674,·)$, $\chi_{705}(676,·)$, $\chi_{705}(166,·)$, $\chi_{705}(554,·)$, $\chi_{705}(299,·)$, $\chi_{705}(314,·)$, $\chi_{705}(571,·)$, $\chi_{705}(316,·)$, $\chi_{705}(61,·)$, $\chi_{705}(194,·)$, $\chi_{705}(451,·)$, $\chi_{705}(196,·)$, $\chi_{705}(74,·)$, $\chi_{705}(331,·)$, $\chi_{705}(209,·)$, $\chi_{705}(526,·)$, $\chi_{705}(601,·)$, $\chi_{705}(346,·)$, $\chi_{705}(479,·)$, $\chi_{705}(224,·)$, $\chi_{705}(59,·)$, $\chi_{705}(614,·)$, $\chi_{705}(361,·)$, $\chi_{705}(106,·)$, $\chi_{705}(494,·)$, $\chi_{705}(239,·)$, $\chi_{705}(241,·)$, $\chi_{705}(629,·)$, $\chi_{705}(119,·)$, $\chi_{705}(121,·)$, $\chi_{705}(661,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{563}a^{44}+\frac{65}{563}a^{43}+\frac{258}{563}a^{42}+\frac{58}{563}a^{41}-\frac{141}{563}a^{40}+\frac{15}{563}a^{39}+\frac{200}{563}a^{38}+\frac{76}{563}a^{37}+\frac{112}{563}a^{36}+\frac{203}{563}a^{35}-\frac{180}{563}a^{34}-\frac{226}{563}a^{33}+\frac{83}{563}a^{32}+\frac{81}{563}a^{31}+\frac{15}{563}a^{30}-\frac{270}{563}a^{29}+\frac{253}{563}a^{28}-\frac{195}{563}a^{27}-\frac{269}{563}a^{26}-\frac{160}{563}a^{25}-\frac{174}{563}a^{24}+\frac{152}{563}a^{23}-\frac{63}{563}a^{22}+\frac{227}{563}a^{21}-\frac{18}{563}a^{20}+\frac{165}{563}a^{19}-\frac{120}{563}a^{18}+\frac{103}{563}a^{17}+\frac{42}{563}a^{16}-\frac{17}{563}a^{15}+\frac{114}{563}a^{14}+\frac{59}{563}a^{13}+\frac{97}{563}a^{12}+\frac{57}{563}a^{11}+\frac{251}{563}a^{10}-\frac{279}{563}a^{9}+\frac{85}{563}a^{8}-\frac{28}{563}a^{7}+\frac{119}{563}a^{6}+\frac{97}{563}a^{5}-\frac{13}{563}a^{4}+\frac{196}{563}a^{3}+\frac{25}{563}a^{2}-\frac{109}{563}a-\frac{49}{563}$, $\frac{1}{45\!\cdots\!93}a^{45}+\frac{31\!\cdots\!16}{45\!\cdots\!93}a^{44}-\frac{12\!\cdots\!68}{45\!\cdots\!93}a^{43}+\frac{20\!\cdots\!62}{45\!\cdots\!93}a^{42}-\frac{20\!\cdots\!76}{45\!\cdots\!93}a^{41}-\frac{43\!\cdots\!43}{45\!\cdots\!93}a^{40}+\frac{83\!\cdots\!17}{45\!\cdots\!93}a^{39}+\frac{85\!\cdots\!39}{45\!\cdots\!93}a^{38}+\frac{13\!\cdots\!86}{45\!\cdots\!93}a^{37}-\frac{11\!\cdots\!40}{45\!\cdots\!93}a^{36}-\frac{15\!\cdots\!21}{45\!\cdots\!93}a^{35}-\frac{16\!\cdots\!19}{45\!\cdots\!93}a^{34}-\frac{60\!\cdots\!92}{45\!\cdots\!93}a^{33}-\frac{13\!\cdots\!30}{45\!\cdots\!93}a^{32}+\frac{58\!\cdots\!84}{45\!\cdots\!93}a^{31}+\frac{10\!\cdots\!59}{45\!\cdots\!93}a^{30}+\frac{55\!\cdots\!48}{45\!\cdots\!93}a^{29}-\frac{22\!\cdots\!35}{45\!\cdots\!93}a^{28}-\frac{20\!\cdots\!95}{45\!\cdots\!93}a^{27}-\frac{18\!\cdots\!97}{45\!\cdots\!93}a^{26}+\frac{10\!\cdots\!53}{45\!\cdots\!93}a^{25}-\frac{13\!\cdots\!46}{45\!\cdots\!93}a^{24}+\frac{12\!\cdots\!06}{45\!\cdots\!93}a^{23}-\frac{42\!\cdots\!23}{45\!\cdots\!93}a^{22}+\frac{18\!\cdots\!00}{45\!\cdots\!93}a^{21}-\frac{21\!\cdots\!12}{45\!\cdots\!93}a^{20}+\frac{89\!\cdots\!93}{45\!\cdots\!93}a^{19}-\frac{20\!\cdots\!89}{45\!\cdots\!93}a^{18}-\frac{21\!\cdots\!52}{45\!\cdots\!93}a^{17}-\frac{99\!\cdots\!57}{45\!\cdots\!93}a^{16}+\frac{11\!\cdots\!28}{45\!\cdots\!93}a^{15}-\frac{13\!\cdots\!76}{45\!\cdots\!93}a^{14}+\frac{92\!\cdots\!94}{45\!\cdots\!93}a^{13}+\frac{88\!\cdots\!88}{45\!\cdots\!93}a^{12}+\frac{51\!\cdots\!16}{45\!\cdots\!93}a^{11}-\frac{16\!\cdots\!07}{45\!\cdots\!93}a^{10}+\frac{71\!\cdots\!76}{45\!\cdots\!93}a^{9}-\frac{51\!\cdots\!32}{45\!\cdots\!93}a^{8}-\frac{93\!\cdots\!74}{45\!\cdots\!93}a^{7}-\frac{11\!\cdots\!50}{45\!\cdots\!93}a^{6}+\frac{22\!\cdots\!30}{45\!\cdots\!93}a^{5}-\frac{12\!\cdots\!27}{45\!\cdots\!93}a^{4}+\frac{16\!\cdots\!69}{45\!\cdots\!93}a^{3}+\frac{68\!\cdots\!95}{45\!\cdots\!93}a^{2}+\frac{24\!\cdots\!97}{45\!\cdots\!93}a-\frac{28\!\cdots\!99}{45\!\cdots\!93}$

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

Class group and class number

not computed

Unit group

Rank:		$22$
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

$C_{46}$ (as 46T1):

A cyclic group of order 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\zeta_{47})^+\)

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	$23^{2}$	R	R	$46$	$46$	$46$	$23^{2}$	$23^{2}$	$23^{2}$	$46$	$23^{2}$	$46$	$46$	$46$	R	$23^{2}$	$46$

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
\(3\)		Deg $46$	$2$	$23$	$23$
\(5\)		Deg $46$	$2$	$23$	$23$
\(47\)	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$
	47.23.22.1	$x^{23} + 47$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$