Number field 30.30.17581401814197409148890873176573567284303576419397.1: \(\Q(\zeta_{77})^+\)

• Label: 30.30.175...397.1
• Degree: $30$
• Signature: $[30, 0]$
• Discriminant: $1.758\times 10^{49}$
• Root discriminant: \(43.80\)
• Ramified primes: $7,11$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

x^30 - x^29 - 30*x^28 + 29*x^27 + 405*x^26 - 377*x^25 - 3250*x^24 + 2901*x^23 + 17249*x^22 - 14697*x^21 - 63734*x^20 + 51590*x^19 + 168035*x^18 - 128611*x^17 - 318629*x^16 + 229651*x^15 + 432159*x^14 - 292608*x^13 - 411241*x^12 + 261924*x^11 + 264472*x^10 - 159873*x^9 - 107406*x^8 + 63143*x^7 + 24051*x^6 - 14609*x^5 - 2010*x^4 + 1562*x^3 - 72*x^2 - 24*x + 1

Invariants

Degree:		$30$
Signature:		$[30, 0]$
Discriminant:		\(17581401814197409148890873176573567284303576419397\) \(\medspace = 7^{25}\cdot 11^{27}\)
Root discriminant:		\(43.80\)
Galois root discriminant:		$7^{5/6}11^{9/10}\approx 43.80279098179992$
Ramified primes:		\(7\), \(11\)
Discriminant root field:		\(\Q(\sqrt{77}) \)
$\card{ \Gal(K/\Q) }$:		$30$
This field is Galois and abelian over $\Q$.
Conductor:		\(77=7\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{77}(64,·)$, $\chi_{77}(1,·)$, $\chi_{77}(67,·)$, $\chi_{77}(4,·)$, $\chi_{77}(6,·)$, $\chi_{77}(71,·)$, $\chi_{77}(9,·)$, $\chi_{77}(10,·)$, $\chi_{77}(76,·)$, $\chi_{77}(13,·)$, $\chi_{77}(15,·)$, $\chi_{77}(16,·)$, $\chi_{77}(17,·)$, $\chi_{77}(19,·)$, $\chi_{77}(23,·)$, $\chi_{77}(24,·)$, $\chi_{77}(25,·)$, $\chi_{77}(68,·)$, $\chi_{77}(36,·)$, $\chi_{77}(37,·)$, $\chi_{77}(40,·)$, $\chi_{77}(41,·)$, $\chi_{77}(52,·)$, $\chi_{77}(53,·)$, $\chi_{77}(54,·)$, $\chi_{77}(73,·)$, $\chi_{77}(58,·)$, $\chi_{77}(60,·)$, $\chi_{77}(61,·)$, $\chi_{77}(62,·)$$\rbrace$
This is not a CM field.

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

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

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

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

Unit group

Rank:		$29$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-1$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136136a^{10}+68068a^{8}-20384a^{6}+3185a^{4}-196a^{2}+2$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{14}-14a^{12}-a^{11}+77a^{10}+11a^{9}-210a^{8}-44a^{7}+294a^{6}+77a^{5}-196a^{4}-55a^{3}+49a^{2}+11a-2$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+a^{21}+12397a^{20}-21a^{19}-40964a^{18}+189a^{17}+94962a^{16}-952a^{15}-155039a^{14}+2940a^{13}+176344a^{12}-5732a^{11}-136059a^{10}+6996a^{9}+67858a^{8}-5103a^{7}-20090a^{6}+1995a^{5}+2989a^{4}-316a^{3}-147a^{2}+3a+1$, $a^{28}-28a^{26}+350a^{24}-2577a^{22}+a^{21}+12419a^{20}-21a^{19}-41173a^{18}+189a^{17}+96084a^{16}-952a^{15}-158779a^{14}+2940a^{13}+184352a^{12}-5734a^{11}-147070a^{10}+7018a^{9}+77296a^{8}-5191a^{7}-24809a^{6}+2149a^{5}+4199a^{4}-426a^{3}-268a^{2}+25a+2$, $a^{28}-28a^{26}+350a^{24}-2577a^{22}+a^{21}+12419a^{20}-21a^{19}-41173a^{18}+189a^{17}+96084a^{16}-952a^{15}-158779a^{14}+2940a^{13}+184352a^{12}-5733a^{11}-147070a^{10}+7007a^{9}+77296a^{8}-5147a^{7}-24809a^{6}+2072a^{5}+4199a^{4}-371a^{3}-268a^{2}+14a+3$, $a^{11}-11a^{9}+43a^{7}-70a^{5}+41a^{3}-4a$, $a^{28}-28a^{26}-a^{25}+350a^{24}+25a^{23}-2576a^{22}-274a^{21}+12397a^{20}+1729a^{19}-40964a^{18}-6936a^{17}+94962a^{16}+18428a^{15}-155040a^{14}-32760a^{13}+176358a^{12}+38467a^{11}-136136a^{10}-28743a^{9}+68068a^{8}+12727a^{7}-20384a^{6}-2926a^{5}+3185a^{4}+264a^{3}-196a^{2}-a+3$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-a^{11}-136136a^{10}+11a^{9}+68068a^{8}-44a^{7}-20384a^{6}+77a^{5}+3185a^{4}-55a^{3}-196a^{2}+11a+2$, $a^{29}-2a^{28}-28a^{27}+56a^{26}+349a^{25}-700a^{24}-2550a^{23}+5150a^{22}+12099a^{21}-24750a^{20}-38984a^{19}+81510a^{18}+86526a^{17}-187680a^{16}-130967a^{15}+302598a^{14}+129695a^{13}-336671a^{12}-75104a^{11}+250084a^{10}+15621a^{9}-116786a^{8}+7710a^{7}+30631a^{6}-5299a^{5}-3459a^{4}+945a^{3}+26a^{2}-20a+1$, $a^{4}-4a^{2}+2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{29}-29a^{27}+377a^{25}-a^{24}-2900a^{23}+25a^{22}+14674a^{21}-274a^{20}-51358a^{19}+1729a^{18}+127261a^{17}-6936a^{16}-224638a^{15}+18427a^{14}+280211a^{13}-32745a^{12}-241281a^{11}+38377a^{10}+137072a^{9}-28468a^{8}-47202a^{7}+12278a^{6}+8245a^{5}-2556a^{4}-436a^{3}+144a^{2}+3a-1$, $a^{5}-5a^{3}+5a+1$, $a^{28}-28a^{26}+349a^{24}-2552a^{22}+a^{21}+12144a^{20}-21a^{19}-39424a^{18}+189a^{17}+88978a^{16}-952a^{15}-139551a^{14}+2939a^{13}+149317a^{12}-5720a^{11}-104599a^{10}+6941a^{9}+44263a^{8}-4982a^{7}-9442a^{6}+1863a^{5}+448a^{4}-250a^{3}+96a^{2}-8a-1$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+3$, $a^{3}-3a$, $a$, $a^{13}-13a^{11}-a^{10}+65a^{9}+10a^{8}-156a^{7}-35a^{6}+182a^{5}+50a^{4}-91a^{3}-25a^{2}+13a+2$, $a^{5}-5a^{3}+5a$, $a^{22}-22a^{20}+209a^{18}-a^{17}-1122a^{16}+17a^{15}+3740a^{14}-119a^{13}-8008a^{12}+442a^{11}+11011a^{10}-935a^{9}-9438a^{8}+1122a^{7}+4719a^{6}-714a^{5}-1210a^{4}+204a^{3}+121a^{2}-17a-2$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1783a^{8}+1394a^{6}-560a^{4}+97a^{2}-4$, $a^{29}-a^{28}-30a^{27}+28a^{26}+406a^{25}-351a^{24}-3275a^{23}+2601a^{22}+17525a^{21}-12673a^{20}-65505a^{19}+42735a^{18}+175350a^{17}-102277a^{16}-338979a^{15}+175388a^{14}+470934a^{13}-215102a^{12}-461720a^{11}+186420a^{10}+308516a^{9}-111473a^{8}-132211a^{7}+44025a^{6}+32521a^{5}-10494a^{4}-3584a^{3}+1195a^{2}+52a-21$ (assuming GRH)
Regulator:		\( 864355592506536.0 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{30}\cdot(2\pi)^{0}\cdot 864355592506536.0 \cdot 1}{2\cdot\sqrt{17581401814197409148890873176573567284303576419397}}\cr\approx \mathstrut & 0.110671442530690 \end{aligned}\] (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

\(\Q(\sqrt{77}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.6.22370117.1, 10.10.39630026842637.1, 15.15.886528337182930278529.1

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	$30$	$30$	$30$	R	R	${\href{/padicField/13.5.0.1}{5} }^{6}$	$15^{2}$	$15^{2}$	${\href{/padicField/23.3.0.1}{3} }^{10}$	${\href{/padicField/29.10.0.1}{10} }^{3}$	$30$	$15^{2}$	${\href{/padicField/41.5.0.1}{5} }^{6}$	${\href{/padicField/43.2.0.1}{2} }^{15}$	$30$	$15^{2}$	$30$

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
\(7\)		Deg $30$	$6$	$5$	$25$
\(11\)		Deg $30$	$10$	$3$	$27$