# Properties

 Label 32.32.3138550867...6448.1 Degree $32$ Signature $[32, 0]$ Discriminant $2^{191}$ Root discriminant $62.63$ Ramified prime $2$ Class number $1$ (GRH) Class group Trivial (GRH) Galois group $C_{32}$ (as 32T33)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -256, 0, 5440, 0, -45696, 0, 201552, 0, -537472, 0, 940576, 0, -1136960, 0, 980628, 0, -615296, 0, 283360, 0, -95680, 0, 23400, 0, -4032, 0, 464, 0, -32, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 980628*x^16 - 1136960*x^14 + 940576*x^12 - 537472*x^10 + 201552*x^8 - 45696*x^6 + 5440*x^4 - 256*x^2 + 2)

gp: K = bnfinit(x^32 - 32*x^30 + 464*x^28 - 4032*x^26 + 23400*x^24 - 95680*x^22 + 283360*x^20 - 615296*x^18 + 980628*x^16 - 1136960*x^14 + 940576*x^12 - 537472*x^10 + 201552*x^8 - 45696*x^6 + 5440*x^4 - 256*x^2 + 2, 1)

## Normalizeddefining polynomial

$$x^{32} - 32 x^{30} + 464 x^{28} - 4032 x^{26} + 23400 x^{24} - 95680 x^{22} + 283360 x^{20} - 615296 x^{18} + 980628 x^{16} - 1136960 x^{14} + 940576 x^{12} - 537472 x^{10} + 201552 x^{8} - 45696 x^{6} + 5440 x^{4} - 256 x^{2} + 2$$

magma: DefiningPolynomial(K);

sage: K.defining_polynomial()

gp: K.pol

## Invariants

 Degree: $32$ magma: Degree(K);  sage: K.degree()  gp: poldegree(K.pol) Signature: $[32, 0]$ magma: Signature(K);  sage: K.signature()  gp: K.sign Discriminant: $$3138550867693340381917894711603833208051177722232017256448=2^{191}$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $62.63$ 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: $2$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$128=2^{7}$$ Dirichlet character group: $\lbrace$$\chi_{128}(1,·), \chi_{128}(5,·), \chi_{128}(9,·), \chi_{128}(13,·), \chi_{128}(17,·), \chi_{128}(21,·), \chi_{128}(25,·), \chi_{128}(29,·), \chi_{128}(33,·), \chi_{128}(37,·), \chi_{128}(41,·), \chi_{128}(45,·), \chi_{128}(49,·), \chi_{128}(53,·), \chi_{128}(57,·), \chi_{128}(61,·), \chi_{128}(65,·), \chi_{128}(69,·), \chi_{128}(73,·), \chi_{128}(77,·), \chi_{128}(81,·), \chi_{128}(85,·), \chi_{128}(89,·), \chi_{128}(93,·), \chi_{128}(97,·), \chi_{128}(101,·), \chi_{128}(105,·), \chi_{128}(109,·), \chi_{128}(113,·), \chi_{128}(117,·), \chi_{128}(121,·), \chi_{128}(125,·)$$\rbrace$ This is not a CM field.

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

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

magma: IntegralBasis(K);

sage: K.integral_basis()

gp: K.zk

## Class group and class number

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

magma: ClassGroup(K);

sage: K.class_group().invariants()

gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);

sage: UK = K.unit_group()

 Rank: $31$ 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: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$7254970870890416000$$ (assuming GRH) 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 $32$ $32$ $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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