Global number field 46.0.2628364239485334411197227624389642263541714240820003570059378902799863859937013225488384.1

• Label: 46.0.26283642394...8384.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-\,2^{46}\cdot 47^{44}$
• 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:		\(-2628364239485334411197227624389642263541714240820003570059378902799863859937013225488384=-\,2^{46}\cdot 47^{44}\)
Root discriminant:		$79.51$
Ramified primes:		$2, 47$
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.

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

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

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

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

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

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	Data not computed
47	Data not computed