Global number field 32.0.4489912604053908534055314729400632754872833954383027744299245049859706934263808.1

• Label: 32.0.44899126040...3808.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $2^{191}\cdot 3^{16}\cdot 7^{16}$
• Root discriminant: $287.00$
• Ramified primes: $2, 3, 7$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{32}$ (as 32T33)

Normalized defining polynomial

\( x^{32} + 672 x^{30} + 204624 x^{28} + 37340352 x^{26} + 4550855400 x^{24} + 390766783680 x^{22} + 24302688046560 x^{20} + 1108202574923136 x^{18} + 37090154929458708 x^{16} + 903064641760733760 x^{14} + 15688695730952383776 x^{12} + 188264348771428605312 x^{10} + 1482581746575000266832 x^{8} + 7058769773166802889856 x^{6} + 17646924432917007224640 x^{4} + 17439313557235630669056 x^{2} + 2861137380483970656642 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(4489912604053908534055314729400632754872833954383027744299245049859706934263808=2^{191}\cdot 3^{16}\cdot 7^{16}\)
Root discriminant:		$287.00$
Ramified primes:		$2, 3, 7$
This field is Galois and abelian over $\Q$.
Conductor:		\(2688=2^{7}\cdot 3\cdot 7\)
Dirichlet character group:		$\lbrace$$\chi_{2688}(1,·)$, $\chi_{2688}(2435,·)$, $\chi_{2688}(2185,·)$, $\chi_{2688}(1931,·)$, $\chi_{2688}(1681,·)$, $\chi_{2688}(1427,·)$, $\chi_{2688}(1177,·)$, $\chi_{2688}(923,·)$, $\chi_{2688}(673,·)$, $\chi_{2688}(419,·)$, $\chi_{2688}(169,·)$, $\chi_{2688}(2603,·)$, $\chi_{2688}(2353,·)$, $\chi_{2688}(2099,·)$, $\chi_{2688}(1849,·)$, $\chi_{2688}(1595,·)$, $\chi_{2688}(1345,·)$, $\chi_{2688}(1091,·)$, $\chi_{2688}(841,·)$, $\chi_{2688}(587,·)$, $\chi_{2688}(337,·)$, $\chi_{2688}(83,·)$, $\chi_{2688}(2521,·)$, $\chi_{2688}(2267,·)$, $\chi_{2688}(2017,·)$, $\chi_{2688}(1763,·)$, $\chi_{2688}(1513,·)$, $\chi_{2688}(1259,·)$, $\chi_{2688}(1009,·)$, $\chi_{2688}(755,·)$, $\chi_{2688}(505,·)$, $\chi_{2688}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{21} a^{2}$, $\frac{1}{21} a^{3}$, $\frac{1}{441} a^{4}$, $\frac{1}{441} a^{5}$, $\frac{1}{9261} a^{6}$, $\frac{1}{9261} a^{7}$, $\frac{1}{194481} a^{8}$, $\frac{1}{194481} a^{9}$, $\frac{1}{4084101} a^{10}$, $\frac{1}{4084101} a^{11}$, $\frac{1}{85766121} a^{12}$, $\frac{1}{85766121} a^{13}$, $\frac{1}{1801088541} a^{14}$, $\frac{1}{1801088541} a^{15}$, $\frac{1}{37822859361} a^{16}$, $\frac{1}{37822859361} a^{17}$, $\frac{1}{794280046581} a^{18}$, $\frac{1}{794280046581} a^{19}$, $\frac{1}{16679880978201} a^{20}$, $\frac{1}{16679880978201} a^{21}$, $\frac{1}{350277500542221} a^{22}$, $\frac{1}{350277500542221} a^{23}$, $\frac{1}{7355827511386641} a^{24}$, $\frac{1}{7355827511386641} a^{25}$, $\frac{1}{154472377739119461} a^{26}$, $\frac{1}{154472377739119461} a^{27}$, $\frac{1}{3243919932521508681} a^{28}$, $\frac{1}{3243919932521508681} a^{29}$, $\frac{1}{68122318582951682301} a^{30}$, $\frac{1}{68122318582951682301} a^{31}$

Class group and class number

Not computed

Unit group

Rank:		$15$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_{32}$ (as 32T33):

A cyclic group of order 32

The 32 conjugacy class representatives for $C_{32}$

Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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	$32$	R	$32$	$32$	${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$	$32$	$16^{2}$	$32$	${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$	$32$	$16^{2}$	$32$	${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$	$32$	$32$

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
3	Data not computed
7	Data not computed