Normalized defining polynomial

\( x^{32} + 1 \) x^32 + 1

Invariants

Degree:		$32$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[0, 16]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(1461501637330902918203684832716283019655932542976\) 1461501637330902918203684832716283019655932542976 \(\medspace = 2^{160}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(32.00\)	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:		$2^{5}\approx 32.0$
Ramified primes:		\(2\) 2	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$32$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(64=2^{6}\)
Dirichlet character group:		$\lbrace$$\chi_{64}(1,·)$, $\chi_{64}(3,·)$, $\chi_{64}(5,·)$, $\chi_{64}(7,·)$, $\chi_{64}(9,·)$, $\chi_{64}(11,·)$, $\chi_{64}(13,·)$, $\chi_{64}(15,·)$, $\chi_{64}(17,·)$, $\chi_{64}(19,·)$, $\chi_{64}(21,·)$, $\chi_{64}(23,·)$, $\chi_{64}(25,·)$, $\chi_{64}(27,·)$, $\chi_{64}(29,·)$, $\chi_{64}(31,·)$, $\chi_{64}(33,·)$, $\chi_{64}(35,·)$, $\chi_{64}(37,·)$, $\chi_{64}(39,·)$, $\chi_{64}(41,·)$, $\chi_{64}(43,·)$, $\chi_{64}(45,·)$, $\chi_{64}(47,·)$, $\chi_{64}(49,·)$, $\chi_{64}(51,·)$, $\chi_{64}(53,·)$, $\chi_{64}(55,·)$, $\chi_{64}(57,·)$, $\chi_{64}(59,·)$, $\chi_{64}(61,·)$, $\chi_{64}(63,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$ 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, a^30, a^31

Class group and class number

$C_{17}$, which has order $17$ (assuming GRH)

Unit group

Rank:		$15$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( a \) a  (order $64$)	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:		$a^{2}+a+1$, $a^{6}+a^{3}+1$, $a^{4}+a^{3}+a^{2}+a+1$, $a^{12}+a^{9}+a^{6}+a^{3}+1$, $a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+1$, $a^{30}-a^{2}+1$, $a^{26}-a^{20}+a^{13}-a^{7}+1$, $a^{27}-a^{18}-a^{13}+a^{9}+a^{4}$, $a^{22}-a^{11}+a$, $a^{31}-a^{30}+a^{29}-a^{28}+a^{27}-a^{26}+a^{25}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}$, $a^{28}-a^{21}+a^{14}-a^{7}+a^{3}$, $a^{26}-a^{6}+1$, $a^{27}-a^{22}+1$, $a^{29}-a^{27}-a^{22}-a^{17}-a^{12}-a^{7}-a^{2}$, $a^{31}-a^{30}+a^{29}-a^{28}+a^{27}-a^{26}+a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-a^{20}+a^{19}-a^{18}+a^{17}-a^{16}+a^{15}-a^{14}+a^{13}-a^{12}+a^{11}-a^{10}+1$ a^2 + a + 1, a^6 + a^3 + 1, a^4 + a^3 + a^2 + a + 1, a^12 + a^9 + a^6 + a^3 + 1, a^18 + a^17 + a^16 + a^15 + a^14 + a^13 + a^12 + a^11 + a^10 + a^9 + a^8 + a^7 + a^6 + a^5 + a^4 + a^3 + a^2 + a + 1, a^30 - a^2 + 1, a^26 - a^20 + a^13 - a^7 + 1, a^27 - a^18 - a^13 + a^9 + a^4, a^22 - a^11 + a, a^31 - a^30 + a^29 - a^28 + a^27 - a^26 + a^25 - a^21 + a^20 - a^19 + a^18 - a^17 + a^16 - a^15 + a^14 - a^10 + a^9 - a^8 + a^7 - a^6 + a^5 - a^4 + a^3, a^28 - a^21 + a^14 - a^7 + a^3, a^26 - a^6 + 1, a^27 - a^22 + 1, a^29 - a^27 - a^22 - a^17 - a^12 - a^7 - a^2, a^31 - a^30 + a^29 - a^28 + a^27 - a^26 + a^25 - a^24 + a^23 - a^22 + a^21 - a^20 + a^19 - a^18 + a^17 - a^16 + a^15 - a^14 + a^13 - a^12 + a^11 - a^10 + 1 (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:		\( 211230625393.46567 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 211230625393.46567 \cdot 17}{64\cdot\sqrt{1461501637330902918203684832716283019655932542976}}\cr\approx \mathstrut & 0.273844531562237 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times C_{16}$ (as 32T32):

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
\(2\) 2		Deg $32$	$32$	$1$	$160$