Number field 32.0.35190667333271321019306672876612934335729762304.1: \(\Q(\zeta_{68})\)

• Label: 32.0.351...304.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $3.519\times 10^{46}$
• Root discriminant: $28.48$
• Ramified primes: $2, 17$
• Class number: $8$ (GRH)
• Class group: $[8]$ (GRH)
• Galois group: $C_2\times C_{16}$ (as 32T32)

Normalized defining polynomial

\( x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(35190667333271321019306672876612934335729762304\)\(\medspace = 2^{32}\cdot 17^{30}\)
Root discriminant:		$28.48$
Ramified primes:		$2, 17$
$|\Gal(K/\Q)|$:		$32$
This field is Galois and abelian over $\Q$.
Conductor:		\(68=2^{2}\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{68}(1,·)$, $\chi_{68}(3,·)$, $\chi_{68}(5,·)$, $\chi_{68}(7,·)$, $\chi_{68}(9,·)$, $\chi_{68}(11,·)$, $\chi_{68}(13,·)$, $\chi_{68}(15,·)$, $\chi_{68}(19,·)$, $\chi_{68}(21,·)$, $\chi_{68}(23,·)$, $\chi_{68}(25,·)$, $\chi_{68}(27,·)$, $\chi_{68}(29,·)$, $\chi_{68}(31,·)$, $\chi_{68}(33,·)$, $\chi_{68}(35,·)$, $\chi_{68}(37,·)$, $\chi_{68}(39,·)$, $\chi_{68}(41,·)$, $\chi_{68}(43,·)$, $\chi_{68}(45,·)$, $\chi_{68}(47,·)$, $\chi_{68}(49,·)$, $\chi_{68}(53,·)$, $\chi_{68}(55,·)$, $\chi_{68}(57,·)$, $\chi_{68}(59,·)$, $\chi_{68}(61,·)$, $\chi_{68}(63,·)$, $\chi_{68}(65,·)$, $\chi_{68}(67,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$

Class group and class number

$C_{8}$, which has order $8$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( a \) (order $68$)
Fundamental units:		\( a^{19} - a^{17} \),  \( a^{4} + 1 \),  \( a^{17} + a \),  \( a^{8} + a^{4} + 1 \),  \( a^{31} + a^{27} + a^{23} - a^{21} + a^{19} - a^{17} + a^{15} - a^{13} - a^{9} - a^{5} - a \),  \( a^{11} - a^{9} + a^{7} \),  \( a^{21} + a^{17} + a^{13} + a^{9} + a^{5} \),  \( a - 1 \),  \( a^{3} - 1 \),  \( a^{9} - 1 \),  \( a^{7} - 1 \),  \( a^{11} - 1 \),  \( a^{13} - 1 \),  \( a^{5} - 1 \),  \( a^{15} - 1 \) (assuming GRH)
Regulator:		\( 36938367173.77557 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 36938367173.77557 \cdot 8}{68\sqrt{35190667333271321019306672876612934335729762304}}\approx 0.136685646313966$ (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{17})\), 4.4.4913.1, 4.0.78608.1, 8.0.6179217664.1, \(\Q(\zeta_{17})^+\), 8.0.105046700288.1, 16.0.11034809241396899282944.1, \(\Q(\zeta_{17})\), \(\Q(\zeta_{68})^+\)

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