Number field 30.30.38731022422755868071217069332926190761458900976553.1: \(\Q(\zeta_{99})^+\)

• Label: 30.30.387...553.1
• Degree: $30$
• Signature: $[30, 0]$
• Discriminant: $3.873\times 10^{49}$
• Root discriminant: $44.97$
• Ramified primes: $3, 11$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

\( x^{30} - 30 x^{28} - x^{27} + 405 x^{26} + 27 x^{25} - 3250 x^{24} - 324 x^{23} + 17250 x^{22} + 2278 x^{21} - 63756 x^{20} - 10416 x^{19} + 168244 x^{18} + 32508 x^{17} - 319752 x^{16} - 70720 x^{15} + 435915 x^{14} + 107592 x^{13} - 419353 x^{12} - 113139 x^{11} + 275835 x^{10} + 79936 x^{9} - 117504 x^{8} - 36027 x^{7} + 29442 x^{6} + 9423 x^{5} - 3555 x^{4} - 1173 x^{3} + 108 x^{2} + 36 x + 1 \)

Invariants

Degree:		$30$
Signature:		$[30, 0]$
Discriminant:		\(38731022422755868071217069332926190761458900976553\)\(\medspace = 3^{45}\cdot 11^{27}\)
Root discriminant:		$44.97$
Ramified primes:		$3, 11$
$|\Gal(K/\Q)|$:		$30$
This field is Galois and abelian over $\Q$.
Conductor:		\(99=3^{2}\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(2,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(68,·)$, $\chi_{99}(65,·)$, $\chi_{99}(8,·)$, $\chi_{99}(74,·)$, $\chi_{99}(98,·)$, $\chi_{99}(16,·)$, $\chi_{99}(17,·)$, $\chi_{99}(82,·)$, $\chi_{99}(83,·)$, $\chi_{99}(25,·)$, $\chi_{99}(91,·)$, $\chi_{99}(29,·)$, $\chi_{99}(31,·)$, $\chi_{99}(32,·)$, $\chi_{99}(97,·)$, $\chi_{99}(34,·)$, $\chi_{99}(35,·)$, $\chi_{99}(37,·)$, $\chi_{99}(41,·)$, $\chi_{99}(49,·)$, $\chi_{99}(50,·)$, $\chi_{99}(58,·)$, $\chi_{99}(95,·)$, $\chi_{99}(70,·)$, $\chi_{99}(62,·)$$\rbrace$
This is not a CM field.

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

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$29$
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:		\( 1458908175124897.5 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{30}\cdot(2\pi)^{0}\cdot 1458908175124897.5 \cdot 1}{2\sqrt{38731022422755868071217069332926190761458900976553}}\approx 0.125854384871860$ (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

A cyclic group of order 30

The 30 conjugacy class representatives for $C_{30}$

Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.6.26198073.1, \(\Q(\zeta_{33})^+\), 15.15.10943023107606534329121.1

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	$15^{2}$	R	$30$	$30$	R	$30$	${\href{/LocalNumberField/17.5.0.1}{5} }^{6}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$	$15^{2}$	$15^{2}$	${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$	$15^{2}$	${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$	$30$	${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$	$30$

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
11	Data not computed