Number field 46.46.445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125.1

• Label: 46.46.445...125.1
• Degree: $46$
• Signature: $[46, 0]$
• Discriminant: $4.453\times 10^{89}$
• Root discriminant: \(88.90\)
• Ramified primes: $5,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - x^45 - 67*x^44 + 62*x^43 + 2017*x^42 - 1712*x^41 - 36151*x^40 + 27871*x^39 + 431244*x^38 - 298794*x^37 - 3629323*x^36 + 2234528*x^35 + 22311838*x^34 - 12065268*x^33 - 102471229*x^32 + 48099584*x^31 + 356957890*x^30 - 143761015*x^29 - 952979825*x^28 + 325518500*x^27 + 1962978291*x^26 - 562037616*x^25 - 3130570732*x^24 + 742002577*x^23 + 3866596078*x^22 - 748176098*x^21 - 3686881837*x^20 + 573064052*x^19 + 2694891598*x^18 - 329962748*x^17 - 1492308292*x^16 + 140457727*x^15 + 615043968*x^14 - 43115073*x^13 - 183867242*x^12 + 9203272*x^11 + 38414233*x^10 - 1297868*x^9 - 5308732*x^8 + 114257*x^7 + 445588*x^6 - 6578*x^5 - 19657*x^4 + 352*x^3 + 342*x^2 - 12*x - 1

Invariants

Degree:		$46$
Signature:		$[46, 0]$
Discriminant:		445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125
Root discriminant:		\(88.90\)
Galois root discriminant:		$5^{1/2}47^{22/23}\approx 88.89615014384839$
Ramified primes:		\(5\), \(47\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(235=5\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{235}(1,·)$, $\chi_{235}(131,·)$, $\chi_{235}(4,·)$, $\chi_{235}(6,·)$, $\chi_{235}(136,·)$, $\chi_{235}(9,·)$, $\chi_{235}(14,·)$, $\chi_{235}(16,·)$, $\chi_{235}(21,·)$, $\chi_{235}(24,·)$, $\chi_{235}(159,·)$, $\chi_{235}(34,·)$, $\chi_{235}(36,·)$, $\chi_{235}(166,·)$, $\chi_{235}(169,·)$, $\chi_{235}(49,·)$, $\chi_{235}(51,·)$, $\chi_{235}(54,·)$, $\chi_{235}(56,·)$, $\chi_{235}(59,·)$, $\chi_{235}(61,·)$, $\chi_{235}(191,·)$, $\chi_{235}(64,·)$, $\chi_{235}(96,·)$, $\chi_{235}(194,·)$, $\chi_{235}(196,·)$, $\chi_{235}(71,·)$, $\chi_{235}(74,·)$, $\chi_{235}(204,·)$, $\chi_{235}(206,·)$, $\chi_{235}(79,·)$, $\chi_{235}(81,·)$, $\chi_{235}(84,·)$, $\chi_{235}(216,·)$, $\chi_{235}(89,·)$, $\chi_{235}(224,·)$, $\chi_{235}(144,·)$, $\chi_{235}(101,·)$, $\chi_{235}(209,·)$, $\chi_{235}(106,·)$, $\chi_{235}(111,·)$, $\chi_{235}(189,·)$, $\chi_{235}(119,·)$, $\chi_{235}(121,·)$, $\chi_{235}(126,·)$, $\chi_{235}(149,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{281}a^{44}+\frac{66}{281}a^{43}+\frac{10}{281}a^{42}+\frac{20}{281}a^{41}+\frac{90}{281}a^{40}+\frac{32}{281}a^{39}+\frac{96}{281}a^{38}+\frac{76}{281}a^{37}+\frac{108}{281}a^{36}+\frac{75}{281}a^{35}+\frac{50}{281}a^{34}+\frac{79}{281}a^{33}+\frac{74}{281}a^{32}+\frac{56}{281}a^{31}-\frac{50}{281}a^{30}+\frac{19}{281}a^{29}+\frac{123}{281}a^{28}-\frac{140}{281}a^{27}+\frac{123}{281}a^{26}-\frac{22}{281}a^{25}-\frac{79}{281}a^{24}+\frac{134}{281}a^{23}+\frac{109}{281}a^{22}-\frac{123}{281}a^{21}-\frac{102}{281}a^{20}+\frac{54}{281}a^{19}+\frac{37}{281}a^{18}-\frac{87}{281}a^{17}-\frac{85}{281}a^{16}+\frac{92}{281}a^{15}-\frac{114}{281}a^{14}-\frac{51}{281}a^{13}+\frac{42}{281}a^{12}+\frac{52}{281}a^{11}+\frac{41}{281}a^{10}-\frac{119}{281}a^{9}+\frac{32}{281}a^{8}-\frac{18}{281}a^{7}-\frac{107}{281}a^{6}+\frac{119}{281}a^{5}-\frac{113}{281}a^{4}-\frac{114}{281}a^{3}+\frac{40}{281}a^{2}-\frac{132}{281}a+\frac{67}{281}$, $\frac{1}{62\!\cdots\!59}a^{45}+\frac{10\!\cdots\!22}{62\!\cdots\!59}a^{44}-\frac{19\!\cdots\!27}{62\!\cdots\!59}a^{43}-\frac{15\!\cdots\!46}{62\!\cdots\!59}a^{42}+\frac{21\!\cdots\!12}{62\!\cdots\!59}a^{41}+\frac{15\!\cdots\!47}{62\!\cdots\!59}a^{40}-\frac{18\!\cdots\!30}{62\!\cdots\!59}a^{39}+\frac{26\!\cdots\!69}{62\!\cdots\!59}a^{38}-\frac{92\!\cdots\!63}{62\!\cdots\!59}a^{37}-\frac{39\!\cdots\!96}{62\!\cdots\!59}a^{36}+\frac{53\!\cdots\!65}{62\!\cdots\!59}a^{35}-\frac{26\!\cdots\!37}{62\!\cdots\!59}a^{34}+\frac{26\!\cdots\!48}{62\!\cdots\!59}a^{33}-\frac{30\!\cdots\!18}{62\!\cdots\!59}a^{32}-\frac{62\!\cdots\!42}{62\!\cdots\!59}a^{31}+\frac{12\!\cdots\!03}{62\!\cdots\!59}a^{30}+\frac{19\!\cdots\!83}{62\!\cdots\!59}a^{29}-\frac{22\!\cdots\!68}{62\!\cdots\!59}a^{28}+\frac{28\!\cdots\!71}{62\!\cdots\!59}a^{27}+\frac{41\!\cdots\!45}{62\!\cdots\!59}a^{26}+\frac{24\!\cdots\!76}{62\!\cdots\!59}a^{25}-\frac{11\!\cdots\!42}{62\!\cdots\!59}a^{24}+\frac{21\!\cdots\!13}{62\!\cdots\!59}a^{23}+\frac{42\!\cdots\!82}{22\!\cdots\!39}a^{22}+\frac{10\!\cdots\!28}{62\!\cdots\!59}a^{21}-\frac{25\!\cdots\!18}{62\!\cdots\!59}a^{20}+\frac{33\!\cdots\!86}{62\!\cdots\!59}a^{19}+\frac{25\!\cdots\!19}{62\!\cdots\!59}a^{18}+\frac{13\!\cdots\!09}{62\!\cdots\!59}a^{17}-\frac{73\!\cdots\!74}{62\!\cdots\!59}a^{16}-\frac{11\!\cdots\!48}{62\!\cdots\!59}a^{15}+\frac{24\!\cdots\!81}{62\!\cdots\!59}a^{14}-\frac{24\!\cdots\!46}{62\!\cdots\!59}a^{13}-\frac{15\!\cdots\!09}{62\!\cdots\!59}a^{12}-\frac{25\!\cdots\!28}{62\!\cdots\!59}a^{11}-\frac{25\!\cdots\!91}{62\!\cdots\!59}a^{10}-\frac{14\!\cdots\!82}{62\!\cdots\!59}a^{9}+\frac{30\!\cdots\!54}{62\!\cdots\!59}a^{8}-\frac{16\!\cdots\!73}{62\!\cdots\!59}a^{7}+\frac{78\!\cdots\!12}{62\!\cdots\!59}a^{6}+\frac{78\!\cdots\!35}{62\!\cdots\!59}a^{5}+\frac{14\!\cdots\!30}{62\!\cdots\!59}a^{4}-\frac{19\!\cdots\!46}{62\!\cdots\!59}a^{3}+\frac{81\!\cdots\!30}{62\!\cdots\!59}a^{2}+\frac{22\!\cdots\!75}{62\!\cdots\!59}a-\frac{26\!\cdots\!20}{62\!\cdots\!59}$

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{47})^+\)

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	$46$	$46$	R	$46$	$23^{2}$	$46$	$46$	$23^{2}$	$46$	$23^{2}$	$23^{2}$	$46$	$23^{2}$	$46$	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
\(5\)		Deg $46$	$2$	$23$	$23$
\(47\)		Deg $46$	$23$	$2$	$44$