Global number field 32.0.2306255574353269294321282113641705264626385702354944.1

• Label: 32.0.23062555743...4944.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $2^{48}\cdot 17^{30}$
• Root discriminant: $40.28$
• Ramified primes: $2, 17$
• Class number: $72$ (GRH)
• Class group: $[3, 24]$ (GRH)
• Galois group: $C_2\times C_{16}$ (as 32T32)

Normalized defining polynomial

\( x^{32} - 2 x^{30} + 4 x^{28} - 8 x^{26} + 16 x^{24} - 32 x^{22} + 64 x^{20} - 128 x^{18} + 256 x^{16} - 512 x^{14} + 1024 x^{12} - 2048 x^{10} + 4096 x^{8} - 8192 x^{6} + 16384 x^{4} - 32768 x^{2} + 65536 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(2306255574353269294321282113641705264626385702354944=2^{48}\cdot 17^{30}\)
Root discriminant:		$40.28$
Ramified primes:		$2, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(136=2^{3}\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{136}(1,·)$, $\chi_{136}(3,·)$, $\chi_{136}(129,·)$, $\chi_{136}(9,·)$, $\chi_{136}(11,·)$, $\chi_{136}(19,·)$, $\chi_{136}(25,·)$, $\chi_{136}(27,·)$, $\chi_{136}(33,·)$, $\chi_{136}(35,·)$, $\chi_{136}(41,·)$, $\chi_{136}(43,·)$, $\chi_{136}(49,·)$, $\chi_{136}(57,·)$, $\chi_{136}(59,·)$, $\chi_{136}(65,·)$, $\chi_{136}(67,·)$, $\chi_{136}(73,·)$, $\chi_{136}(75,·)$, $\chi_{136}(81,·)$, $\chi_{136}(83,·)$, $\chi_{136}(89,·)$, $\chi_{136}(91,·)$, $\chi_{136}(97,·)$, $\chi_{136}(99,·)$, $\chi_{136}(131,·)$, $\chi_{136}(105,·)$, $\chi_{136}(107,·)$, $\chi_{136}(113,·)$, $\chi_{136}(115,·)$, $\chi_{136}(121,·)$, $\chi_{136}(123,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$

Class group and class number

$C_{3}\times C_{24}$, which has order $72$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( -\frac{1}{4096} a^{24} \) (order $34$)
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:		\( 598124168304.8442 \) (assuming GRH)

Galois group

$C_2\times C_{16}$ (as 32T32):

An abelian group of order 32

The 32 conjugacy class representatives for $C_2\times C_{16}$

Character table for $C_2\times C_{16}$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{-2}, \sqrt{17})\), 4.4.4913.1, 4.0.314432.2, 8.0.98867482624.1, \(\Q(\zeta_{17})^+\), 8.0.1680747204608.1, 16.0.2824911165797606216433664.1, \(\Q(\zeta_{17})\), 16.16.48023489818559305679372288.1

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	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$	R	${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	$16^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$	${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
17	Data not computed