Number field 46.0.2628364239485334411197227624389642263541714240820003570059378902799863859937013225488384.1

• Label: 46.0.262...384.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-2.628\times 10^{87}$
• Root discriminant: \(79.51\)
• Ramified primes: $2,47$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 + 45*x^44 + 946*x^42 + 12341*x^40 + 111930*x^38 + 749398*x^36 + 3838380*x^34 + 15380937*x^32 + 48903492*x^30 + 124403620*x^28 + 254186856*x^26 + 417225900*x^24 + 548354040*x^22 + 573166440*x^20 + 471435600*x^18 + 300540195*x^16 + 145422675*x^14 + 51895935*x^12 + 13123110*x^10 + 2220075*x^8 + 230230*x^6 + 12650*x^4 + 276*x^2 + 1

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-2628364239485334411197227624389642263541714240820003570059378902799863859937013225488384
Root discriminant:		\(79.51\)
Galois root discriminant:		$2\cdot 47^{22/23}\approx 79.51113386386908$
Ramified primes:		\(2\), \(47\)
Discriminant root field:		\(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(188=2^{2}\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{188}(1,·)$, $\chi_{188}(3,·)$, $\chi_{188}(7,·)$, $\chi_{188}(9,·)$, $\chi_{188}(143,·)$, $\chi_{188}(115,·)$, $\chi_{188}(145,·)$, $\chi_{188}(131,·)$, $\chi_{188}(21,·)$, $\chi_{188}(153,·)$, $\chi_{188}(25,·)$, $\chi_{188}(155,·)$, $\chi_{188}(157,·)$, $\chi_{188}(159,·)$, $\chi_{188}(27,·)$, $\chi_{188}(37,·)$, $\chi_{188}(177,·)$, $\chi_{188}(169,·)$, $\chi_{188}(173,·)$, $\chi_{188}(175,·)$, $\chi_{188}(49,·)$, $\chi_{188}(51,·)$, $\chi_{188}(53,·)$, $\chi_{188}(55,·)$, $\chi_{188}(59,·)$, $\chi_{188}(61,·)$, $\chi_{188}(63,·)$, $\chi_{188}(65,·)$, $\chi_{188}(71,·)$, $\chi_{188}(75,·)$, $\chi_{188}(79,·)$, $\chi_{188}(81,·)$, $\chi_{188}(83,·)$, $\chi_{188}(89,·)$, $\chi_{188}(95,·)$, $\chi_{188}(97,·)$, $\chi_{188}(101,·)$, $\chi_{188}(165,·)$, $\chi_{188}(17,·)$, $\chi_{188}(103,·)$, $\chi_{188}(111,·)$, $\chi_{188}(147,·)$, $\chi_{188}(119,·)$, $\chi_{188}(121,·)$, $\chi_{188}(183,·)$, $\chi_{188}(149,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $a^{44}$, $a^{45}$

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$22$
Torsion generator:		\( -a^{45} - 44 a^{43} - 903 a^{41} - 11480 a^{39} - 101270 a^{37} - 658008 a^{35} - 3262623 a^{33} - 12620256 a^{31} - 38608020 a^{29} - 94143280 a^{27} - 183579396 a^{25} - 286097760 a^{23} - 354817320 a^{21} - 347373600 a^{19} - 265182525 a^{17} - 155117520 a^{15} - 67863915 a^{13} - 21474180 a^{11} - 4686825 a^{9} - 657800 a^{7} - 53130 a^{5} - 2024 a^{3} - 23 a \)  (order $4$)
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

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{47})^+\)

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