Global number field 16.0.121029087867608368152576.1

• Label: 16.0.12102908786...2576.1
• Degree: $16$
• Signature: $[0, 8]$
• Discriminant: $2^{64}\cdot 3^{8}$
• Root discriminant: $27.71$
• Ramified primes: $2, 3$
• Class number: $18$ (GRH)
• Class group: $[3, 6]$ (GRH)
• Galois group: $C_8\times C_2$ (as 16T5)

Normalized defining polynomial

\( x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 1 \)

Invariants

Degree:		$16$
Signature:		$[0, 8]$
Discriminant:		\(121029087867608368152576=2^{64}\cdot 3^{8}\)
Root discriminant:		$27.71$
Ramified primes:		$2, 3$
This field is Galois and abelian over $\Q$.
Conductor:		\(96=2^{5}\cdot 3\)
Dirichlet character group:		$\lbrace$$\chi_{96}(1,·)$, $\chi_{96}(67,·)$, $\chi_{96}(5,·)$, $\chi_{96}(71,·)$, $\chi_{96}(73,·)$, $\chi_{96}(77,·)$, $\chi_{96}(19,·)$, $\chi_{96}(23,·)$, $\chi_{96}(25,·)$, $\chi_{96}(91,·)$, $\chi_{96}(29,·)$, $\chi_{96}(95,·)$, $\chi_{96}(43,·)$, $\chi_{96}(47,·)$, $\chi_{96}(49,·)$, $\chi_{96}(53,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (assuming GRH)

Unit group

Rank:		$7$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( a^{4} + 4 a^{2} + 2 \),  \( a^{10} + 10 a^{8} + 35 a^{6} + 50 a^{4} + 25 a^{2} + 2 \),  \( a^{6} + 6 a^{4} + 10 a^{2} + 4 \),  \( a^{7} + 7 a^{5} + 14 a^{3} + 7 a \),  \( a^{7} + 6 a^{5} + 10 a^{3} + 4 a \),  \( a^{13} + 13 a^{11} + 65 a^{9} + 156 a^{7} + 182 a^{5} + 92 a^{3} + 16 a \),  \( a^{13} + 13 a^{11} + 65 a^{9} + 156 a^{7} + 182 a^{5} + 91 a^{3} + 13 a \) (assuming GRH)
Regulator:		\( 5982.15532136 \) (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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{16})^+\), 4.4.18432.1, \(\Q(\zeta_{48})^+\), 8.0.2147483648.1, 8.0.173946175488.1

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	${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$	${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$	${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$	${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$	${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$	${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$	${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$	${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
3	Data not computed