Number field 28.28.14603047886206093768209337615200705673567789821.1: \(\Q(\zeta_{87})^+\)

• Label: 28.28.146...821.1
• Degree: $28$
• Signature: $[28, 0]$
• Discriminant: $1.460\times 10^{46}$
• Root discriminant: \(44.54\)
• Ramified primes: $3,29$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{28}$ (as 28T1)

Normalized defining polynomial

x^28 - x^27 - 28*x^26 + 28*x^25 + 349*x^24 - 349*x^23 - 2551*x^22 + 2551*x^21 + 12123*x^20 - 12123*x^19 - 39236*x^18 + 39236*x^17 + 88045*x^16 - 88045*x^15 - 136763*x^14 + 136763*x^13 + 144247*x^12 - 144247*x^11 - 99295*x^10 + 99295*x^9 + 41703*x^8 - 41703*x^7 - 9569*x^6 + 9569*x^5 + 987*x^4 - 987*x^3 - 28*x^2 + 28*x + 1

Invariants

Degree:		$28$
Signature:		$[28, 0]$
Discriminant:		\(14603047886206093768209337615200705673567789821\) \(\medspace = 3^{14}\cdot 29^{27}\)
Root discriminant:		\(44.54\)
Galois root discriminant:		$3^{1/2}29^{27/28}\approx 44.53793994501929$
Ramified primes:		\(3\), \(29\)
Discriminant root field:		\(\Q(\sqrt{29}) \)
$\card{ \Gal(K/\Q) }$:		$28$
This field is Galois and abelian over $\Q$.
Conductor:		\(87=3\cdot 29\)
Dirichlet character group:		$\lbrace$$\chi_{87}(64,·)$, $\chi_{87}(1,·)$, $\chi_{87}(2,·)$, $\chi_{87}(67,·)$, $\chi_{87}(4,·)$, $\chi_{87}(68,·)$, $\chi_{87}(7,·)$, $\chi_{87}(8,·)$, $\chi_{87}(11,·)$, $\chi_{87}(13,·)$, $\chi_{87}(14,·)$, $\chi_{87}(77,·)$, $\chi_{87}(16,·)$, $\chi_{87}(17,·)$, $\chi_{87}(82,·)$, $\chi_{87}(22,·)$, $\chi_{87}(25,·)$, $\chi_{87}(26,·)$, $\chi_{87}(28,·)$, $\chi_{87}(32,·)$, $\chi_{87}(34,·)$, $\chi_{87}(41,·)$, $\chi_{87}(44,·)$, $\chi_{87}(47,·)$, $\chi_{87}(49,·)$, $\chi_{87}(50,·)$, $\chi_{87}(52,·)$, $\chi_{87}(56,·)$$\rbrace$
This is not a CM field.

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

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

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

Class group and class number

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

Unit group

Rank:		$27$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+a^{7}+4719a^{6}-7a^{5}-1210a^{4}+14a^{3}+121a^{2}-7a-2$, $a^{27}-28a^{25}+a^{24}+349a^{23}-25a^{22}-2551a^{21}+274a^{20}+12124a^{19}-1728a^{18}-39255a^{17}+6917a^{16}+88197a^{15}-18277a^{14}-137428a^{13}+32111a^{12}+145976a^{11}-36840a^{10}-102012a^{9}+26355a^{8}+44211a^{7}-10780a^{6}-10823a^{5}+2148a^{4}+1272a^{3}-143a^{2}-46a-1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{6}-6a^{4}+a^{3}+9a^{2}-3a-1$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a+1$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+a^{7}+4719a^{6}-7a^{5}-1210a^{4}+14a^{3}+121a^{2}-7a-3$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{25}-25a^{23}+274a^{21}-1729a^{19}+6936a^{17}+a^{16}-18428a^{15}-16a^{14}+32761a^{13}+103a^{12}-38480a^{11}-340a^{10}+28808a^{9}+606a^{8}-12883a^{7}-560a^{6}+3108a^{5}+232a^{4}-356a^{3}-32a^{2}+17a+1$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+3$, $a^{15}-15a^{13}+90a^{11}-274a^{9}+441a^{7}-351a^{5}+111a^{3}-9a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}+a^{8}-5148a^{7}-8a^{6}+2079a^{5}+20a^{4}-385a^{3}-16a^{2}+21a+2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}+a^{2}-27a-2$, $a^{4}-4a^{2}+2$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-2$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}+a^{6}-3289a^{5}-6a^{4}+506a^{3}+9a^{2}-23a-2$, $a^{17}-17a^{15}+119a^{13}+a^{12}-442a^{11}-12a^{10}+935a^{9}+54a^{8}-1122a^{7}-112a^{6}+714a^{5}+105a^{4}-204a^{3}-36a^{2}+17a+2$, $a^{27}-28a^{25}+a^{24}+349a^{23}-25a^{22}-2551a^{21}+274a^{20}+12123a^{19}-1728a^{18}-39236a^{17}+6917a^{16}+88045a^{15}-18277a^{14}-136763a^{13}+32111a^{12}+144247a^{11}-36841a^{10}-99295a^{9}+26365a^{8}+41703a^{7}-10815a^{6}-9569a^{5}+2198a^{4}+987a^{3}-168a^{2}-28a+1$, $a^{27}-28a^{25}+a^{24}+349a^{23}-25a^{22}-2551a^{21}+274a^{20}+12123a^{19}-1728a^{18}-39236a^{17}+6917a^{16}+88045a^{15}-18277a^{14}-136763a^{13}+32111a^{12}+144247a^{11}-36841a^{10}-99295a^{9}+26365a^{8}+41703a^{7}-10815a^{6}-9568a^{5}+2198a^{4}+982a^{3}-168a^{2}-22a+1$, $a^{15}+a^{14}-15a^{13}-14a^{12}+90a^{11}+77a^{10}-275a^{9}-210a^{8}+450a^{7}+294a^{6}-378a^{5}-196a^{4}+140a^{3}+49a^{2}-15a-2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+a^{9}+19305a^{8}-9a^{7}-8008a^{6}+28a^{5}+1716a^{4}-35a^{3}-144a^{2}+14a+2$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}-1254a^{5}+284a^{3}-16a$, $a^{24}-25a^{22}+274a^{20}-1729a^{18}+6936a^{16}+a^{15}-18428a^{14}-15a^{13}+32760a^{12}+89a^{11}-38467a^{10}-264a^{9}+28743a^{8}+406a^{7}-12727a^{6}-301a^{5}+2926a^{4}+85a^{3}-265a^{2}-4a+3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+a^{5}+1716a^{4}-5a^{3}-144a^{2}+5a+2$ (assuming GRH)
Regulator:		\( 114482618238342.14 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot 114482618238342.14 \cdot 1}{2\cdot\sqrt{14603047886206093768209337615200705673567789821}}\cr\approx \mathstrut & 0.127153313363501 \end{aligned}\] (assuming GRH)

Galois group

$C_{28}$ (as 28T1):

A cyclic group of order 28

The 28 conjugacy class representatives for $C_{28}$

Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.219501.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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	$28$	R	${\href{/padicField/5.7.0.1}{7} }^{4}$	${\href{/padicField/7.7.0.1}{7} }^{4}$	$28$	${\href{/padicField/13.14.0.1}{14} }^{2}$	${\href{/padicField/17.4.0.1}{4} }^{7}$	$28$	${\href{/padicField/23.14.0.1}{14} }^{2}$	R	$28$	$28$	${\href{/padicField/41.4.0.1}{4} }^{7}$	$28$	$28$	${\href{/padicField/53.14.0.1}{14} }^{2}$	${\href{/padicField/59.2.0.1}{2} }^{14}$

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
\(3\)		Deg $28$	$2$	$14$	$14$
\(29\)		Deg $28$	$28$	$1$	$27$