Number field 46.46.247442564620359691917237351409538478343164666147650139214878531054838634357304717043417009323245568.1

• Label: 46.46.247...568.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $2.474\times 10^{98}$
• Root discriminant: \(137.72\)
• Ramified primes: $2,3,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - 2*x^45 - 112*x^44 + 218*x^43 + 5713*x^42 - 10796*x^41 - 175759*x^40 + 321578*x^39 + 3640569*x^38 - 6429108*x^37 - 53679062*x^36 + 91168168*x^35 + 580933861*x^34 - 944995322*x^33 - 4690600862*x^32 + 7273130902*x^31 + 28447305226*x^30 - 41812973042*x^29 - 129501143256*x^28 + 179268136002*x^27 + 439325593874*x^26 - 568431783454*x^25 - 1095481009606*x^24 + 1313118990890*x^23 + 1966263727274*x^22 - 2161106685334*x^21 - 2466848961868*x^20 + 2457189744590*x^19 + 2078976791930*x^18 - 1853404680970*x^17 - 1116883023433*x^16 + 880987194236*x^15 + 358010044796*x^14 - 248497252516*x^13 - 63604513444*x^12 + 39246867980*x^11 + 5926504964*x^10 - 3353727748*x^9 - 285501034*x^8 + 152340728*x^7 + 6709600*x^6 - 3466424*x^5 - 67372*x^4 + 33776*x^3 + 264*x^2 - 96*x + 1

Invariants

Degree:		$46$
Signature:		$[46, 0]$
Discriminant:		247442564620359691917237351409538478343164666147650139214878531054838634357304717043417009323245568
Root discriminant:		\(137.72\)
Galois root discriminant:		$2\cdot 3^{1/2}47^{22/23}\approx 137.71732361963154$
Ramified primes:		\(2\), \(3\), \(47\)
Discriminant root field:		\(\Q(\sqrt{3}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(564=2^{2}\cdot 3\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{564}(1,·)$, $\chi_{564}(131,·)$, $\chi_{564}(263,·)$, $\chi_{564}(397,·)$, $\chi_{564}(143,·)$, $\chi_{564}(529,·)$, $\chi_{564}(277,·)$, $\chi_{564}(25,·)$, $\chi_{564}(155,·)$, $\chi_{564}(541,·)$, $\chi_{564}(289,·)$, $\chi_{564}(37,·)$, $\chi_{564}(551,·)$, $\chi_{564}(553,·)$, $\chi_{564}(95,·)$, $\chi_{564}(299,·)$, $\chi_{564}(157,·)$, $\chi_{564}(385,·)$, $\chi_{564}(49,·)$, $\chi_{564}(59,·)$, $\chi_{564}(61,·)$, $\chi_{564}(191,·)$, $\chi_{564}(71,·)$, $\chi_{564}(457,·)$, $\chi_{564}(119,·)$, $\chi_{564}(205,·)$, $\chi_{564}(335,·)$, $\chi_{564}(337,·)$, $\chi_{564}(83,·)$, $\chi_{564}(215,·)$, $\chi_{564}(347,·)$, $\chi_{564}(479,·)$, $\chi_{564}(97,·)$, $\chi_{564}(455,·)$, $\chi_{564}(145,·)$, $\chi_{564}(361,·)$, $\chi_{564}(431,·)$, $\chi_{564}(491,·)$, $\chi_{564}(239,·)$, $\chi_{564}(241,·)$, $\chi_{564}(371,·)$, $\chi_{564}(169,·)$, $\chi_{564}(121,·)$, $\chi_{564}(251,·)$, $\chi_{564}(253,·)$, $\chi_{564}(383,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{659}a^{44}+\frac{326}{659}a^{43}+\frac{149}{659}a^{42}-\frac{324}{659}a^{41}-\frac{12}{659}a^{40}-\frac{63}{659}a^{39}+\frac{185}{659}a^{38}+\frac{236}{659}a^{37}+\frac{259}{659}a^{36}-\frac{234}{659}a^{35}-\frac{21}{659}a^{34}+\frac{286}{659}a^{33}-\frac{44}{659}a^{32}-\frac{134}{659}a^{31}-\frac{166}{659}a^{30}+\frac{216}{659}a^{29}+\frac{215}{659}a^{28}+\frac{226}{659}a^{27}-\frac{59}{659}a^{26}+\frac{153}{659}a^{25}+\frac{279}{659}a^{24}+\frac{23}{659}a^{23}-\frac{69}{659}a^{22}+\frac{149}{659}a^{21}+\frac{167}{659}a^{20}+\frac{209}{659}a^{19}+\frac{266}{659}a^{18}+\frac{54}{659}a^{17}+\frac{104}{659}a^{16}+\frac{316}{659}a^{15}+\frac{44}{659}a^{14}+\frac{9}{659}a^{13}+\frac{329}{659}a^{12}+\frac{314}{659}a^{11}+\frac{236}{659}a^{10}+\frac{289}{659}a^{9}+\frac{287}{659}a^{8}+\frac{193}{659}a^{7}-\frac{31}{659}a^{6}-\frac{156}{659}a^{5}-\frac{289}{659}a^{4}+\frac{322}{659}a^{3}+\frac{83}{659}a^{2}+\frac{19}{659}a+\frac{210}{659}$, $\frac{1}{71\!\cdots\!69}a^{45}-\frac{31\!\cdots\!21}{71\!\cdots\!69}a^{44}+\frac{22\!\cdots\!70}{71\!\cdots\!69}a^{43}+\frac{32\!\cdots\!07}{71\!\cdots\!69}a^{42}-\frac{22\!\cdots\!83}{71\!\cdots\!69}a^{41}-\frac{13\!\cdots\!65}{71\!\cdots\!69}a^{40}-\frac{32\!\cdots\!71}{71\!\cdots\!69}a^{39}+\frac{18\!\cdots\!88}{71\!\cdots\!69}a^{38}+\frac{31\!\cdots\!71}{71\!\cdots\!69}a^{37}+\frac{12\!\cdots\!44}{71\!\cdots\!69}a^{36}-\frac{82\!\cdots\!08}{71\!\cdots\!69}a^{35}+\frac{93\!\cdots\!17}{71\!\cdots\!69}a^{34}-\frac{30\!\cdots\!10}{71\!\cdots\!69}a^{33}+\frac{13\!\cdots\!53}{71\!\cdots\!69}a^{32}-\frac{20\!\cdots\!76}{71\!\cdots\!69}a^{31}+\frac{26\!\cdots\!88}{71\!\cdots\!69}a^{30}-\frac{20\!\cdots\!08}{71\!\cdots\!69}a^{29}+\frac{86\!\cdots\!57}{71\!\cdots\!69}a^{28}-\frac{18\!\cdots\!15}{71\!\cdots\!69}a^{27}+\frac{19\!\cdots\!52}{71\!\cdots\!69}a^{26}+\frac{13\!\cdots\!25}{71\!\cdots\!69}a^{25}+\frac{39\!\cdots\!85}{71\!\cdots\!69}a^{24}-\frac{62\!\cdots\!08}{71\!\cdots\!69}a^{23}+\frac{26\!\cdots\!19}{71\!\cdots\!69}a^{22}+\frac{67\!\cdots\!67}{71\!\cdots\!69}a^{21}-\frac{26\!\cdots\!98}{71\!\cdots\!69}a^{20}-\frac{24\!\cdots\!30}{71\!\cdots\!69}a^{19}+\frac{28\!\cdots\!31}{71\!\cdots\!69}a^{18}+\frac{21\!\cdots\!84}{71\!\cdots\!69}a^{17}+\frac{15\!\cdots\!10}{71\!\cdots\!69}a^{16}+\frac{53\!\cdots\!53}{71\!\cdots\!69}a^{15}-\frac{27\!\cdots\!47}{71\!\cdots\!69}a^{14}-\frac{57\!\cdots\!44}{71\!\cdots\!69}a^{13}-\frac{11\!\cdots\!07}{71\!\cdots\!69}a^{12}+\frac{20\!\cdots\!03}{71\!\cdots\!69}a^{11}+\frac{16\!\cdots\!33}{71\!\cdots\!69}a^{10}+\frac{32\!\cdots\!49}{71\!\cdots\!69}a^{9}+\frac{35\!\cdots\!07}{71\!\cdots\!69}a^{8}+\frac{10\!\cdots\!43}{71\!\cdots\!69}a^{7}-\frac{49\!\cdots\!34}{71\!\cdots\!69}a^{6}+\frac{25\!\cdots\!49}{71\!\cdots\!69}a^{5}+\frac{68\!\cdots\!83}{71\!\cdots\!69}a^{4}-\frac{33\!\cdots\!42}{71\!\cdots\!69}a^{3}+\frac{16\!\cdots\!11}{71\!\cdots\!69}a^{2}-\frac{19\!\cdots\!54}{71\!\cdots\!69}a+\frac{41\!\cdots\!42}{71\!\cdots\!69}$

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

Class group and class number

not computed

Unit group

Rank:		$45$
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

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{47})^+\)

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