Global number field 16.0.7205759403792793600000000.1

• Label: 16.0.72057594037...0000.1
• Degree: $16$
• Signature: $[0, 8]$
• Discriminant: $2^{64}\cdot 5^{8}$
• Root discriminant: $35.78$
• Ramified primes: $2, 5$
• Class number: $34$ (GRH)
• Class group: $[34]$ (GRH)
• Galois group: $C_8\times C_2$ (as 16T5)

Normalized defining polynomial

\( x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2209 \)

Invariants

Degree:		$16$
Signature:		$[0, 8]$
Discriminant:		\(7205759403792793600000000=2^{64}\cdot 5^{8}\)
Root discriminant:		$35.78$
Ramified primes:		$2, 5$
This field is Galois and abelian over $\Q$.
Conductor:		\(160=2^{5}\cdot 5\)
Dirichlet character group:		$\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(139,·)$, $\chi_{160}(141,·)$, $\chi_{160}(79,·)$, $\chi_{160}(81,·)$, $\chi_{160}(19,·)$, $\chi_{160}(21,·)$, $\chi_{160}(159,·)$, $\chi_{160}(99,·)$, $\chi_{160}(101,·)$, $\chi_{160}(39,·)$, $\chi_{160}(41,·)$, $\chi_{160}(119,·)$, $\chi_{160}(121,·)$, $\chi_{160}(59,·)$, $\chi_{160}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{21} a^{8} + \frac{8}{21} a^{6} - \frac{1}{21} a^{4} - \frac{5}{21} a^{2} + \frac{2}{21}$, $\frac{1}{987} a^{9} + \frac{386}{987} a^{7} - \frac{295}{987} a^{5} + \frac{373}{987} a^{3} - \frac{313}{987} a$, $\frac{1}{987} a^{10} + \frac{10}{987} a^{8} - \frac{114}{329} a^{6} - \frac{34}{141} a^{4} - \frac{407}{987} a^{2} + \frac{5}{21}$, $\frac{1}{987} a^{11} - \frac{254}{987} a^{7} - \frac{83}{329} a^{5} - \frac{9}{47} a^{3} + \frac{404}{987} a$, $\frac{1}{987} a^{12} - \frac{19}{987} a^{8} - \frac{49}{141} a^{6} - \frac{424}{987} a^{4} + \frac{72}{329} a^{2} + \frac{10}{21}$, $\frac{1}{987} a^{13} + \frac{82}{987} a^{7} - \frac{107}{987} a^{5} + \frac{394}{987} a^{3} + \frac{445}{987} a$, $\frac{1}{987} a^{14} - \frac{4}{329} a^{8} + \frac{128}{987} a^{6} + \frac{488}{987} a^{4} - \frac{24}{329} a^{2} - \frac{4}{21}$, $\frac{1}{987} a^{15} - \frac{25}{141} a^{7} - \frac{13}{141} a^{5} + \frac{152}{329} a^{3} + \frac{4}{987} a$

Class group and class number

$C_{34}$, which has order $34$ (assuming GRH)

Unit group

Rank:		$7$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 15753.9498624 \) (assuming GRH)

Galois group

$C_2\times C_8$ (as 16T5):

An abelian group of order 16

The 16 conjugacy class representatives for $C_8\times C_2$

Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\zeta_{16})^+\), 4.0.51200.2, 8.0.10485760000.2, 8.0.1342177280000.1, \(\Q(\zeta_{32})^+\)

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	${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$	R	${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$	${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$	${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$	${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$	${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$	${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$	${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2	Data not computed
5	Data not computed