Number field 46.0.779912673387430559616650036621334842027644229827742389590869715354976917232287630948371314414901652178727.1

• Label: 46.0.779...727.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-7.799\times 10^{104}$
• Root discriminant: \(190.66\)
• Ramified primes: $23,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - 21*x^45 + 303*x^44 - 3199*x^43 + 28462*x^42 - 216844*x^41 + 1482276*x^40 - 9149385*x^39 + 52159796*x^38 - 275416018*x^37 + 1364962019*x^36 - 6356196961*x^35 + 28058790010*x^34 - 117422665453*x^33 + 468880651758*x^32 - 1785260085988*x^31 + 6514636802898*x^30 - 22754700540704*x^29 + 76404337315353*x^28 - 246165239380046*x^27 + 763986254214168*x^26 - 2278363058201293*x^25 + 6553379006345079*x^24 - 18122639402283948*x^23 + 48369398641755245*x^22 - 124082565817885020*x^21 + 307258023227043408*x^20 - 730443630601059097*x^19 + 1675623338370133488*x^18 - 3682257303813422851*x^17 + 7802310077530656188*x^16 - 15781930061220186535*x^15 + 30744365804979004797*x^14 - 56866181492428688346*x^13 + 101154460314253792185*x^12 - 169422446704123589990*x^11 + 272498861417357563948*x^10 - 407198743058120303990*x^9 + 583869749598099518224*x^8 - 760308418085795942035*x^7 + 951731184738366402827*x^6 - 1036802148342691785802*x^5 + 1096694466642564711209*x^4 - 920087778256872115878*x^3 + 778468706568920738062*x^2 - 399355828906278303580*x + 243039256589954114401

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-779912673387430559616650036621334842027644229827742389590869715354976917232287630948371314414901652178727
Root discriminant:		\(190.66\)
Galois root discriminant:		$23^{1/2}47^{22/23}\approx 190.6610011193404$
Ramified primes:		\(23\), \(47\)
Discriminant root field:		\(\Q(\sqrt{-23}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(1081=23\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{1081}(896,·)$, $\chi_{1081}(1,·)$, $\chi_{1081}(643,·)$, $\chi_{1081}(645,·)$, $\chi_{1081}(390,·)$, $\chi_{1081}(392,·)$, $\chi_{1081}(1036,·)$, $\chi_{1081}(277,·)$, $\chi_{1081}(24,·)$, $\chi_{1081}(921,·)$, $\chi_{1081}(666,·)$, $\chi_{1081}(413,·)$, $\chi_{1081}(162,·)$, $\chi_{1081}(1059,·)$, $\chi_{1081}(806,·)$, $\chi_{1081}(551,·)$, $\chi_{1081}(553,·)$, $\chi_{1081}(298,·)$, $\chi_{1081}(300,·)$, $\chi_{1081}(942,·)$, $\chi_{1081}(944,·)$, $\chi_{1081}(183,·)$, $\chi_{1081}(827,·)$, $\chi_{1081}(576,·)$, $\chi_{1081}(68,·)$, $\chi_{1081}(965,·)$, $\chi_{1081}(967,·)$, $\chi_{1081}(712,·)$, $\chi_{1081}(714,·)$, $\chi_{1081}(459,·)$, $\chi_{1081}(206,·)$, $\chi_{1081}(850,·)$, $\chi_{1081}(852,·)$, $\chi_{1081}(346,·)$, $\chi_{1081}(988,·)$, $\chi_{1081}(990,·)$, $\chi_{1081}(737,·)$, $\chi_{1081}(482,·)$, $\chi_{1081}(484,·)$, $\chi_{1081}(873,·)$, $\chi_{1081}(620,·)$, $\chi_{1081}(1011,·)$, $\chi_{1081}(758,·)$, $\chi_{1081}(760,·)$, $\chi_{1081}(507,·)$, $\chi_{1081}(252,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $\frac{1}{14\!\cdots\!79}a^{45}+\frac{33\!\cdots\!02}{14\!\cdots\!79}a^{44}-\frac{58\!\cdots\!69}{14\!\cdots\!79}a^{43}-\frac{56\!\cdots\!09}{14\!\cdots\!79}a^{42}-\frac{56\!\cdots\!64}{14\!\cdots\!79}a^{41}+\frac{22\!\cdots\!09}{14\!\cdots\!79}a^{40}-\frac{43\!\cdots\!72}{14\!\cdots\!79}a^{39}-\frac{56\!\cdots\!43}{14\!\cdots\!79}a^{38}+\frac{64\!\cdots\!09}{14\!\cdots\!79}a^{37}-\frac{19\!\cdots\!78}{14\!\cdots\!79}a^{36}+\frac{26\!\cdots\!48}{14\!\cdots\!79}a^{35}+\frac{50\!\cdots\!39}{14\!\cdots\!79}a^{34}-\frac{41\!\cdots\!71}{14\!\cdots\!79}a^{33}+\frac{42\!\cdots\!40}{14\!\cdots\!79}a^{32}-\frac{93\!\cdots\!84}{14\!\cdots\!79}a^{31}-\frac{13\!\cdots\!99}{14\!\cdots\!79}a^{30}-\frac{69\!\cdots\!41}{14\!\cdots\!79}a^{29}-\frac{62\!\cdots\!32}{14\!\cdots\!79}a^{28}+\frac{23\!\cdots\!40}{14\!\cdots\!79}a^{27}+\frac{11\!\cdots\!15}{14\!\cdots\!79}a^{26}+\frac{46\!\cdots\!27}{14\!\cdots\!79}a^{25}+\frac{87\!\cdots\!89}{14\!\cdots\!79}a^{24}+\frac{15\!\cdots\!54}{14\!\cdots\!79}a^{23}-\frac{69\!\cdots\!10}{14\!\cdots\!79}a^{22}-\frac{59\!\cdots\!78}{14\!\cdots\!79}a^{21}-\frac{32\!\cdots\!03}{14\!\cdots\!79}a^{20}-\frac{67\!\cdots\!36}{14\!\cdots\!79}a^{19}-\frac{15\!\cdots\!84}{14\!\cdots\!79}a^{18}-\frac{26\!\cdots\!93}{14\!\cdots\!79}a^{17}-\frac{67\!\cdots\!78}{14\!\cdots\!79}a^{16}-\frac{52\!\cdots\!98}{14\!\cdots\!79}a^{15}+\frac{54\!\cdots\!57}{14\!\cdots\!79}a^{14}+\frac{55\!\cdots\!17}{14\!\cdots\!79}a^{13}-\frac{69\!\cdots\!76}{14\!\cdots\!79}a^{12}-\frac{59\!\cdots\!63}{14\!\cdots\!79}a^{11}-\frac{35\!\cdots\!61}{14\!\cdots\!79}a^{10}-\frac{41\!\cdots\!44}{14\!\cdots\!79}a^{9}-\frac{73\!\cdots\!58}{14\!\cdots\!79}a^{8}-\frac{23\!\cdots\!36}{14\!\cdots\!79}a^{7}-\frac{49\!\cdots\!44}{14\!\cdots\!79}a^{6}-\frac{38\!\cdots\!45}{14\!\cdots\!79}a^{5}+\frac{22\!\cdots\!25}{14\!\cdots\!79}a^{4}-\frac{31\!\cdots\!95}{14\!\cdots\!79}a^{3}-\frac{54\!\cdots\!59}{14\!\cdots\!79}a^{2}-\frac{25\!\cdots\!39}{14\!\cdots\!79}a+\frac{89\!\cdots\!78}{14\!\cdots\!79}$

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 $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{-23}) \), \(\Q(\zeta_{47})^+\)

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

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