Number field 30.30.5950661074415937716058277355262049126611998411687341.1: \(\Q(\zeta_{61})^+\)

• Label: 30.30.595...341.1
• Degree: $30$
• Signature: $[30, 0]$
• Discriminant: $5.951\times 10^{51}$
• Root discriminant: $53.19$
• Ramified prime: $61$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

\( x^{30} - x^{29} - 29 x^{28} + 28 x^{27} + 378 x^{26} - 351 x^{25} - 2925 x^{24} + 2600 x^{23} + 14950 x^{22} - 12650 x^{21} - 53130 x^{20} + 42504 x^{19} + 134596 x^{18} - 100947 x^{17} - 245157 x^{16} + 170544 x^{15} + 319770 x^{14} - 203490 x^{13} - 293930 x^{12} + 167960 x^{11} + 184756 x^{10} - 92378 x^{9} - 75582 x^{8} + 31824 x^{7} + 18564 x^{6} - 6188 x^{5} - 2380 x^{4} + 560 x^{3} + 120 x^{2} - 15 x - 1 \)

Invariants

Degree:		$30$
Signature:		$[30, 0]$
Discriminant:		\(5950661074415937716058277355262049126611998411687341\)\(\medspace = 61^{29}\)
Root discriminant:		$53.19$
Ramified primes:		$61$
$|\Gal(K/\Q)|$:		$30$
This field is Galois and abelian over $\Q$.
Conductor:		\(61\)
Dirichlet character group:		$\lbrace$$\chi_{61}(1,·)$, $\chi_{61}(3,·)$, $\chi_{61}(4,·)$, $\chi_{61}(5,·)$, $\chi_{61}(9,·)$, $\chi_{61}(12,·)$, $\chi_{61}(13,·)$, $\chi_{61}(14,·)$, $\chi_{61}(15,·)$, $\chi_{61}(16,·)$, $\chi_{61}(19,·)$, $\chi_{61}(20,·)$, $\chi_{61}(22,·)$, $\chi_{61}(25,·)$, $\chi_{61}(27,·)$, $\chi_{61}(34,·)$, $\chi_{61}(36,·)$, $\chi_{61}(39,·)$, $\chi_{61}(41,·)$, $\chi_{61}(42,·)$, $\chi_{61}(45,·)$, $\chi_{61}(46,·)$, $\chi_{61}(47,·)$, $\chi_{61}(48,·)$, $\chi_{61}(49,·)$, $\chi_{61}(52,·)$, $\chi_{61}(56,·)$, $\chi_{61}(57,·)$, $\chi_{61}(58,·)$, $\chi_{61}(60,·)$$\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:		\( 17190292874679612 \) (assuming GRH)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{30}\cdot(2\pi)^{0}\cdot 17190292874679612 \cdot 1}{2\sqrt{5950661074415937716058277355262049126611998411687341}}\approx 0.119638385183071$ (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

\(\Q(\sqrt{61}) \), 3.3.3721.1, 5.5.13845841.1, 6.6.844596301.1, 10.10.11694146092834141.1, 15.15.9876832533361318095112441.1

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	$30$	${\href{/LocalNumberField/3.5.0.1}{5} }^{6}$	$15^{2}$	$30$	${\href{/LocalNumberField/11.2.0.1}{2} }^{15}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{10}$	$30$	$15^{2}$	${\href{/LocalNumberField/23.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{5}$	$30$	${\href{/LocalNumberField/37.10.0.1}{10} }^{3}$	${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$	$30$	${\href{/LocalNumberField/47.3.0.1}{3} }^{10}$	${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$	$30$

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
61	Data not computed