Normalized defining polynomial

\( x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} + \cdots + 1 \) x^28 - x^27 + x^26 - x^25 + x^24 - x^23 + x^22 - x^21 + x^20 - x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1

Invariants

Degree:		$28$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 14]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(3053134545970524535745336759489912159909\) 3053134545970524535745336759489912159909 \(\medspace = 29^{27}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(25.71\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$29^{27/28}\approx 25.713991616408272$
Ramified primes:		\(29\) 29	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{29}) \)
$\card{ \Gal(K/\Q) }$:		$28$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(29\)
Dirichlet character group:		$\lbrace$$\chi_{29}(1,·)$, $\chi_{29}(2,·)$, $\chi_{29}(3,·)$, $\chi_{29}(4,·)$, $\chi_{29}(5,·)$, $\chi_{29}(6,·)$, $\chi_{29}(7,·)$, $\chi_{29}(8,·)$, $\chi_{29}(9,·)$, $\chi_{29}(10,·)$, $\chi_{29}(11,·)$, $\chi_{29}(12,·)$, $\chi_{29}(13,·)$, $\chi_{29}(14,·)$, $\chi_{29}(15,·)$, $\chi_{29}(16,·)$, $\chi_{29}(17,·)$, $\chi_{29}(18,·)$, $\chi_{29}(19,·)$, $\chi_{29}(20,·)$, $\chi_{29}(21,·)$, $\chi_{29}(22,·)$, $\chi_{29}(23,·)$, $\chi_{29}(24,·)$, $\chi_{29}(25,·)$, $\chi_{29}(26,·)$, $\chi_{29}(27,·)$, $\chi_{29}(28,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{8192}$

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

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH)

Unit group

Rank:		$13$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( a \) a  (order $58$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$a-1$, $a^{2}+1$, $a^{3}-1$, $a^{2}-a+1$, $a^{6}+1$, $a^{4}+1$, $a^{16}-a^{3}$, $a^{4}-a^{3}+a^{2}-a+1$, $a^{20}+a^{10}+1$, $a^{20}-a^{11}+a^{2}$, $a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}$, $a^{22}-a^{15}+a^{8}-a$, $a^{17}+a^{3}$ a - 1, a^2 + 1, a^3 - 1, a^2 - a + 1, a^6 + 1, a^4 + 1, a^16 - a^3, a^4 - a^3 + a^2 - a + 1, a^20 + a^10 + 1, a^20 - a^11 + a^2, a^27 + a^25 + a^23 + a^21 + a^19 + a^17 + a^15 + a^13 + a^11, a^22 - a^15 + a^8 - a, a^17 + a^3 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 487075979.1876791 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 8}{58\cdot\sqrt{3053134545970524535745336759489912159909}}\cr\approx \mathstrut & 0.181720486605872 \end{aligned}\] (assuming GRH)

Galois group

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

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

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
\(29\) 29		Deg $28$	$28$	$1$	$27$