Global number field 46.0.2579429343542351596390826632891502881531178841973772779681657383147657295402368825407289691573406771522895573153759625216.1

• Label: 46.0.25794293435...5216.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-\,2^{46}\cdot 23^{23}\cdot 47^{45}$
• Root discriminant: $414.61$
• Ramified primes: $2, 23, 47$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

\( x^{46} + 1081 x^{44} + 546986 x^{42} + 172126549 x^{40} + 37747752490 x^{38} + 6126999482734 x^{36} + 763035594118044 x^{34} + 74586729325038801 x^{32} + 5806290005918405124 x^{30} + 363147787212265162580 x^{28} + 18330129929665578854984 x^{26} + 747369388495910192405484 x^{24} + 24556422764865620607608760 x^{22} + 646578389451371250297173640 x^{20} + 13519366324892307960759085200 x^{18} + 221548615649172696706939508715 x^{16} + 2794371248994404013303656384115 x^{14} + 26464339475770532125993451637795 x^{12} + 181904310419664117371771081372430 x^{10} + 865071250679981610809362473444075 x^{8} + 2652885168751943606482044918561830 x^{6} + 4693566067791900226852848702070930 x^{4} + 3925527983971407462458746187186596 x^{2} + 981381995992851865614686546796649 \)

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		\(-2579429343542351596390826632891502881531178841973772779681657383147657295402368825407289691573406771522895573153759625216=-\,2^{46}\cdot 23^{23}\cdot 47^{45}\)
Root discriminant:		$414.61$
Ramified primes:		$2, 23, 47$
This field is Galois and abelian over $\Q$.
Conductor:		\(4324=2^{2}\cdot 23\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{4324}(1,·)$, $\chi_{4324}(3587,·)$, $\chi_{4324}(3589,·)$, $\chi_{4324}(1381,·)$, $\chi_{4324}(4233,·)$, $\chi_{4324}(1933,·)$, $\chi_{4324}(275,·)$, $\chi_{4324}(277,·)$, $\chi_{4324}(919,·)$, $\chi_{4324}(921,·)$, $\chi_{4324}(3679,·)$, $\chi_{4324}(1565,·)$, $\chi_{4324}(645,·)$, $\chi_{4324}(2851,·)$, $\chi_{4324}(553,·)$, $\chi_{4324}(1195,·)$, $\chi_{4324}(1841,·)$, $\chi_{4324}(91,·)$, $\chi_{4324}(2483,·)$, $\chi_{4324}(3129,·)$, $\chi_{4324}(2207,·)$, $\chi_{4324}(1473,·)$, $\chi_{4324}(2943,·)$, $\chi_{4324}(2117,·)$, $\chi_{4324}(2759,·)$, $\chi_{4324}(3403,·)$, $\chi_{4324}(3405,·)$, $\chi_{4324}(4047,·)$, $\chi_{4324}(1105,·)$, $\chi_{4324}(2391,·)$, $\chi_{4324}(1103,·)$, $\chi_{4324}(735,·)$, $\chi_{4324}(737,·)$, $\chi_{4324}(4323,·)$, $\chi_{4324}(3771,·)$, $\chi_{4324}(4049,·)$, $\chi_{4324}(2025,·)$, $\chi_{4324}(2667,·)$, $\chi_{4324}(2669,·)$, $\chi_{4324}(367,·)$, $\chi_{4324}(3219,·)$, $\chi_{4324}(3957,·)$, $\chi_{4324}(1655,·)$, $\chi_{4324}(1657,·)$, $\chi_{4324}(2299,·)$, $\chi_{4324}(3221,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{23} a^{2}$, $\frac{1}{23} a^{3}$, $\frac{1}{529} a^{4}$, $\frac{1}{529} a^{5}$, $\frac{1}{12167} a^{6}$, $\frac{1}{12167} a^{7}$, $\frac{1}{279841} a^{8}$, $\frac{1}{279841} a^{9}$, $\frac{1}{6436343} a^{10}$, $\frac{1}{6436343} a^{11}$, $\frac{1}{148035889} a^{12}$, $\frac{1}{148035889} a^{13}$, $\frac{1}{3404825447} a^{14}$, $\frac{1}{3404825447} a^{15}$, $\frac{1}{78310985281} a^{16}$, $\frac{1}{78310985281} a^{17}$, $\frac{1}{1801152661463} a^{18}$, $\frac{1}{1801152661463} a^{19}$, $\frac{1}{41426511213649} a^{20}$, $\frac{1}{41426511213649} a^{21}$, $\frac{1}{952809757913927} a^{22}$, $\frac{1}{952809757913927} a^{23}$, $\frac{1}{21914624432020321} a^{24}$, $\frac{1}{21914624432020321} a^{25}$, $\frac{1}{504036361936467383} a^{26}$, $\frac{1}{504036361936467383} a^{27}$, $\frac{1}{11592836324538749809} a^{28}$, $\frac{1}{11592836324538749809} a^{29}$, $\frac{1}{266635235464391245607} a^{30}$, $\frac{1}{266635235464391245607} a^{31}$, $\frac{1}{6132610415680998648961} a^{32}$, $\frac{1}{6132610415680998648961} a^{33}$, $\frac{1}{141050039560662968926103} a^{34}$, $\frac{1}{141050039560662968926103} a^{35}$, $\frac{1}{3244150909895248285300369} a^{36}$, $\frac{1}{3244150909895248285300369} a^{37}$, $\frac{1}{74615470927590710561908487} a^{38}$, $\frac{1}{74615470927590710561908487} a^{39}$, $\frac{1}{1716155831334586342923895201} a^{40}$, $\frac{1}{1716155831334586342923895201} a^{41}$, $\frac{1}{39471584120695485887249589623} a^{42}$, $\frac{1}{39471584120695485887249589623} a^{43}$, $\frac{1}{907846434775996175406740561329} a^{44}$, $\frac{1}{907846434775996175406740561329} a^{45}$

Class group and class number

Not computed

Unit group

Rank:		$22$
Torsion generator:		\( -1 \) (order $2$)
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

\(\Q(\sqrt{-1081}) \), \(\Q(\zeta_{47})^+\)

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

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
23	Data not computed
47	Data not computed