Number field 32.0.352701833122210710593389803720131611763844129.1: \(\Q(\zeta_{51})\)

• Label: 32.0.352...129.1
• Degree: $32$
• Signature: $[0, 16]$
• Discriminant: $3.527\times 10^{44}$
• Root discriminant: $24.67$
• Ramified primes: $3, 17$
• Class number: $5$ (GRH)
• Class group: $[5]$ (GRH)
• Galois group: $C_2\times C_{16}$ (as 32T32)

Normalized defining polynomial

\( x^{32} - x^{31} + x^{29} - x^{28} + x^{26} - x^{25} + x^{23} - x^{22} + x^{20} - x^{19} + x^{17} - x^{16} + x^{15} - x^{13} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)

Invariants

Degree:		$32$
Signature:		$[0, 16]$
Discriminant:		\(352701833122210710593389803720131611763844129\)\(\medspace = 3^{16}\cdot 17^{30}\)
Root discriminant:		$24.67$
Ramified primes:		$3, 17$
$|\Gal(K/\Q)|$:		$32$
This field is Galois and abelian over $\Q$.
Conductor:		\(51=3\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{51}(1,·)$, $\chi_{51}(2,·)$, $\chi_{51}(4,·)$, $\chi_{51}(5,·)$, $\chi_{51}(7,·)$, $\chi_{51}(8,·)$, $\chi_{51}(10,·)$, $\chi_{51}(11,·)$, $\chi_{51}(13,·)$, $\chi_{51}(14,·)$, $\chi_{51}(16,·)$, $\chi_{51}(19,·)$, $\chi_{51}(20,·)$, $\chi_{51}(22,·)$, $\chi_{51}(23,·)$, $\chi_{51}(25,·)$, $\chi_{51}(26,·)$, $\chi_{51}(28,·)$, $\chi_{51}(29,·)$, $\chi_{51}(31,·)$, $\chi_{51}(32,·)$, $\chi_{51}(35,·)$, $\chi_{51}(37,·)$, $\chi_{51}(38,·)$, $\chi_{51}(40,·)$, $\chi_{51}(41,·)$, $\chi_{51}(43,·)$, $\chi_{51}(44,·)$, $\chi_{51}(46,·)$, $\chi_{51}(47,·)$, $\chi_{51}(49,·)$, $\chi_{51}(50,·)$$\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

$C_{5}$, which has order $5$ (assuming GRH)

Unit group

Rank:		$15$
Torsion generator:		\( -a \) (order $102$)
Fundamental units:		\( a^{3} + 1 \),  \( a^{6} + 1 \),  \( a^{21} - a^{19} + a^{6} - a^{2} \),  \( a^{19} - a^{18} + a^{2} - 1 \),  \( a^{6} + a^{3} + 1 \),  \( a^{18} + a^{15} + a^{12} + a^{9} + a^{6} + a^{3} + 1 \),  \( a^{27} + a^{3} \),  \( a - 1 \),  \( a^{2} - 1 \),  \( a^{4} - 1 \),  \( a^{8} - 1 \),  \( a^{5} - 1 \),  \( a^{7} - 1 \),  \( a^{11} - 1 \),  \( a^{14} - 1 \) (assuming GRH)
Regulator:		\( 9977645145.96466 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 9977645145.96466 \cdot 5}{102\sqrt{352701833122210710593389803720131611763844129}}\approx 0.153663925012584$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{-3}, \sqrt{17})\), 4.4.4913.1, 4.0.44217.1, 8.0.1955143089.1, \(\Q(\zeta_{17})^+\), 8.0.33237432513.1, 16.0.1104726920056229495169.1, \(\Q(\zeta_{17})\), \(\Q(\zeta_{51})^+\)

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	${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$	R	$16^{2}$	$16^{2}$	$16^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$	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
3	Data not computed
17	Data not computed