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Properties

Label 32.32.4789048565...0000.1
Degree $32$
Signature $[32, 0]$
Discriminant $2^{191}\cdot 5^{16}$
Root discriminant $140.04$
Ramified primes $2, 5$
Class number Not computed
Class group Not computed
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![305175781250, 0, -7812500000000, 0, 33203125000000, 0, -55781250000000, 0, 49207031250000, 0, -26243750000000, 0, 9185312500000, 0, -2220625000000, 0, 383057812500, 0, -48070000000, 0, 4427500000, 0, -299000000, 0, 14625000, 0, -504000, 0, 11600, 0, -160, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 305175781250)
 
gp: K = bnfinit(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 305175781250, 1)
 

Normalized defining polynomial

\( x^{32} - 160 x^{30} + 11600 x^{28} - 504000 x^{26} + 14625000 x^{24} - 299000000 x^{22} + 4427500000 x^{20} - 48070000000 x^{18} + 383057812500 x^{16} - 2220625000000 x^{14} + 9185312500000 x^{12} - 26243750000000 x^{10} + 49207031250000 x^{8} - 55781250000000 x^{6} + 33203125000000 x^{4} - 7812500000000 x^{2} + 305175781250 \)

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:  \(478904856520590268236983445984471619880855975682375680000000000000000=2^{191}\cdot 5^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $140.04$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(640=2^{7}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{640}(1,·)$, $\chi_{640}(389,·)$, $\chi_{640}(521,·)$, $\chi_{640}(269,·)$, $\chi_{640}(401,·)$, $\chi_{640}(149,·)$, $\chi_{640}(281,·)$, $\chi_{640}(29,·)$, $\chi_{640}(161,·)$, $\chi_{640}(549,·)$, $\chi_{640}(41,·)$, $\chi_{640}(429,·)$, $\chi_{640}(561,·)$, $\chi_{640}(309,·)$, $\chi_{640}(441,·)$, $\chi_{640}(189,·)$, $\chi_{640}(321,·)$, $\chi_{640}(69,·)$, $\chi_{640}(201,·)$, $\chi_{640}(589,·)$, $\chi_{640}(81,·)$, $\chi_{640}(469,·)$, $\chi_{640}(601,·)$, $\chi_{640}(349,·)$, $\chi_{640}(481,·)$, $\chi_{640}(229,·)$, $\chi_{640}(361,·)$, $\chi_{640}(109,·)$, $\chi_{640}(241,·)$, $\chi_{640}(629,·)$, $\chi_{640}(121,·)$, $\chi_{640}(509,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{78125} a^{14}$, $\frac{1}{78125} a^{15}$, $\frac{1}{390625} a^{16}$, $\frac{1}{390625} a^{17}$, $\frac{1}{1953125} a^{18}$, $\frac{1}{1953125} a^{19}$, $\frac{1}{9765625} a^{20}$, $\frac{1}{9765625} a^{21}$, $\frac{1}{48828125} a^{22}$, $\frac{1}{48828125} a^{23}$, $\frac{1}{244140625} a^{24}$, $\frac{1}{244140625} a^{25}$, $\frac{1}{1220703125} a^{26}$, $\frac{1}{1220703125} a^{27}$, $\frac{1}{6103515625} a^{28}$, $\frac{1}{6103515625} a^{29}$, $\frac{1}{30517578125} a^{30}$, $\frac{1}{30517578125} a^{31}$

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:  $31$
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:  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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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 R $32$ R $16^{2}$ $32$ $32$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{4}$ $32$ $32$

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
5Data not computed