# Properties

 Label 32.0.14421556453...4128.1 Degree $32$ Signature $[0, 16]$ Discriminant $2^{191}\cdot 11^{16}$ Root discriminant $207.72$ Ramified primes $2, 11$ Class number Not computed Class group Not computed Galois group $C_{32}$ (as 32T33)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![91899459727144322, 0, 1069375531370406656, 0, 2065839094692831040, 0, 1577549854129070976, 0, 632556516184870992, 0, 153347034226635392, 0, 24396119081510176, 0, 2680892206759360, 0, 210206320757268, 0, 11990378367616, 0, 501989524960, 0, 15409359680, 0, 342599400, 0, 5366592, 0, 56144, 0, 352, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 352*x^30 + 56144*x^28 + 5366592*x^26 + 342599400*x^24 + 15409359680*x^22 + 501989524960*x^20 + 11990378367616*x^18 + 210206320757268*x^16 + 2680892206759360*x^14 + 24396119081510176*x^12 + 153347034226635392*x^10 + 632556516184870992*x^8 + 1577549854129070976*x^6 + 2065839094692831040*x^4 + 1069375531370406656*x^2 + 91899459727144322)

gp: K = bnfinit(x^32 + 352*x^30 + 56144*x^28 + 5366592*x^26 + 342599400*x^24 + 15409359680*x^22 + 501989524960*x^20 + 11990378367616*x^18 + 210206320757268*x^16 + 2680892206759360*x^14 + 24396119081510176*x^12 + 153347034226635392*x^10 + 632556516184870992*x^8 + 1577549854129070976*x^6 + 2065839094692831040*x^4 + 1069375531370406656*x^2 + 91899459727144322, 1)

## Normalizeddefining polynomial

$$x^{32} + 352 x^{30} + 56144 x^{28} + 5366592 x^{26} + 342599400 x^{24} + 15409359680 x^{22} + 501989524960 x^{20} + 11990378367616 x^{18} + 210206320757268 x^{16} + 2680892206759360 x^{14} + 24396119081510176 x^{12} + 153347034226635392 x^{10} + 632556516184870992 x^{8} + 1577549854129070976 x^{6} + 2065839094692831040 x^{4} + 1069375531370406656 x^{2} + 91899459727144322$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $32$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[0, 16]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$144215564533589000876246801170130951941346253827716612240069988132090544128=2^{191}\cdot 11^{16}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $207.72$ magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));  sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 11$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$1408=2^{7}\cdot 11$$ Dirichlet character group: $\lbrace$$\chi_{1408}(1,·), \chi_{1408}(901,·), \chi_{1408}(265,·), \chi_{1408}(1165,·), \chi_{1408}(529,·), \chi_{1408}(21,·), \chi_{1408}(793,·), \chi_{1408}(285,·), \chi_{1408}(1057,·), \chi_{1408}(549,·), \chi_{1408}(1321,·), \chi_{1408}(813,·), \chi_{1408}(177,·), \chi_{1408}(1077,·), \chi_{1408}(441,·), \chi_{1408}(1341,·), \chi_{1408}(705,·), \chi_{1408}(197,·), \chi_{1408}(969,·), \chi_{1408}(461,·), \chi_{1408}(1233,·), \chi_{1408}(725,·), \chi_{1408}(89,·), \chi_{1408}(989,·), \chi_{1408}(353,·), \chi_{1408}(1253,·), \chi_{1408}(617,·), \chi_{1408}(109,·), \chi_{1408}(881,·), \chi_{1408}(373,·), \chi_{1408}(1145,·), \chi_{1408}(637,·)$$\rbrace$ This is a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $\frac{1}{11} a^{2}$, $\frac{1}{11} a^{3}$, $\frac{1}{121} a^{4}$, $\frac{1}{121} a^{5}$, $\frac{1}{1331} a^{6}$, $\frac{1}{1331} a^{7}$, $\frac{1}{14641} a^{8}$, $\frac{1}{14641} a^{9}$, $\frac{1}{161051} a^{10}$, $\frac{1}{161051} a^{11}$, $\frac{1}{1771561} a^{12}$, $\frac{1}{1771561} a^{13}$, $\frac{1}{19487171} a^{14}$, $\frac{1}{19487171} a^{15}$, $\frac{1}{214358881} a^{16}$, $\frac{1}{214358881} a^{17}$, $\frac{1}{2357947691} a^{18}$, $\frac{1}{2357947691} a^{19}$, $\frac{1}{25937424601} a^{20}$, $\frac{1}{25937424601} a^{21}$, $\frac{1}{285311670611} a^{22}$, $\frac{1}{285311670611} a^{23}$, $\frac{1}{3138428376721} a^{24}$, $\frac{1}{3138428376721} a^{25}$, $\frac{1}{34522712143931} a^{26}$, $\frac{1}{34522712143931} a^{27}$, $\frac{1}{379749833583241} a^{28}$, $\frac{1}{379749833583241} a^{29}$, $\frac{1}{4177248169415651} a^{30}$, $\frac{1}{4177248169415651} a^{31}$

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

Not computed

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $15$ magma: UnitRank(K);  sage: UK.rank()  gp: K.fu Torsion generator: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);  sage: UK.torsion_generator()  gp: K.tu[2] Fundamental units: Not computed magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: Not computed magma: Regulator(K);  sage: K.regulator()  gp: K.reg

## Galois group

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

 A cyclic group of order 32 The 32 conjugacy class representatives for $C_{32}$ Character table for $C_{32}$ is not computed

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R $32$ $32$ $16^{2}$ R $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

magma: idealfactors := Factorization(p*Integers(K)); // get the data

magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:

gp: idealfactors = idealprimedec(K, p); \\ get the data

gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
11Data not computed