Properties

Label 32.0.90528804883...7088.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{191}\cdot 19^{16}$
Root discriminant $272.99$
Ramified primes $2, 19$
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![576882827135242335362, 0, 3886368519647948364544, 0, 4346596370658889618240, 0, 1921653132291298568064, 0, 446098048567622881872, 0, 62610252430543562368, 0, 5766733776497433376, 0, 366883004866811840, 0, 16654557457769748, 0, 549995705519744, 0, 13330920840160, 0, 236913152320, 0, 3049511400, 0, 27655488, 0, 167504, 0, 608, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 608*x^30 + 167504*x^28 + 27655488*x^26 + 3049511400*x^24 + 236913152320*x^22 + 13330920840160*x^20 + 549995705519744*x^18 + 16654557457769748*x^16 + 366883004866811840*x^14 + 5766733776497433376*x^12 + 62610252430543562368*x^10 + 446098048567622881872*x^8 + 1921653132291298568064*x^6 + 4346596370658889618240*x^4 + 3886368519647948364544*x^2 + 576882827135242335362)
 
gp: K = bnfinit(x^32 + 608*x^30 + 167504*x^28 + 27655488*x^26 + 3049511400*x^24 + 236913152320*x^22 + 13330920840160*x^20 + 549995705519744*x^18 + 16654557457769748*x^16 + 366883004866811840*x^14 + 5766733776497433376*x^12 + 62610252430543562368*x^10 + 446098048567622881872*x^8 + 1921653132291298568064*x^6 + 4346596370658889618240*x^4 + 3886368519647948364544*x^2 + 576882827135242335362, 1)
 

Normalized defining polynomial

\( x^{32} + 608 x^{30} + 167504 x^{28} + 27655488 x^{26} + 3049511400 x^{24} + 236913152320 x^{22} + 13330920840160 x^{20} + 549995705519744 x^{18} + 16654557457769748 x^{16} + 366883004866811840 x^{14} + 5766733776497433376 x^{12} + 62610252430543562368 x^{10} + 446098048567622881872 x^{8} + 1921653132291298568064 x^{6} + 4346596370658889618240 x^{4} + 3886368519647948364544 x^{2} + 576882827135242335362 \)

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:  \(905288048831351058796666807211863041216387224344298280390835989733155786457088=2^{191}\cdot 19^{16}\)
magma: Discriminant(K);
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $272.99$
magma: Abs(Discriminant(K))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 19$
magma: PrimeDivisors(Discriminant(K));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2432=2^{7}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2432}(1,·)$, $\chi_{2432}(645,·)$, $\chi_{2432}(1673,·)$, $\chi_{2432}(2317,·)$, $\chi_{2432}(913,·)$, $\chi_{2432}(1557,·)$, $\chi_{2432}(153,·)$, $\chi_{2432}(797,·)$, $\chi_{2432}(1825,·)$, $\chi_{2432}(37,·)$, $\chi_{2432}(1065,·)$, $\chi_{2432}(1709,·)$, $\chi_{2432}(305,·)$, $\chi_{2432}(949,·)$, $\chi_{2432}(1977,·)$, $\chi_{2432}(189,·)$, $\chi_{2432}(1217,·)$, $\chi_{2432}(1861,·)$, $\chi_{2432}(457,·)$, $\chi_{2432}(1101,·)$, $\chi_{2432}(2129,·)$, $\chi_{2432}(341,·)$, $\chi_{2432}(1369,·)$, $\chi_{2432}(2013,·)$, $\chi_{2432}(609,·)$, $\chi_{2432}(1253,·)$, $\chi_{2432}(2281,·)$, $\chi_{2432}(493,·)$, $\chi_{2432}(1521,·)$, $\chi_{2432}(2165,·)$, $\chi_{2432}(761,·)$, $\chi_{2432}(1405,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{19} a^{2}$, $\frac{1}{19} a^{3}$, $\frac{1}{361} a^{4}$, $\frac{1}{361} a^{5}$, $\frac{1}{6859} a^{6}$, $\frac{1}{6859} a^{7}$, $\frac{1}{130321} a^{8}$, $\frac{1}{130321} a^{9}$, $\frac{1}{2476099} a^{10}$, $\frac{1}{2476099} a^{11}$, $\frac{1}{47045881} a^{12}$, $\frac{1}{47045881} a^{13}$, $\frac{1}{893871739} a^{14}$, $\frac{1}{893871739} a^{15}$, $\frac{1}{16983563041} a^{16}$, $\frac{1}{16983563041} a^{17}$, $\frac{1}{322687697779} a^{18}$, $\frac{1}{322687697779} a^{19}$, $\frac{1}{6131066257801} a^{20}$, $\frac{1}{6131066257801} a^{21}$, $\frac{1}{116490258898219} a^{22}$, $\frac{1}{116490258898219} a^{23}$, $\frac{1}{2213314919066161} a^{24}$, $\frac{1}{2213314919066161} a^{25}$, $\frac{1}{42052983462257059} a^{26}$, $\frac{1}{42052983462257059} a^{27}$, $\frac{1}{799006685782884121} a^{28}$, $\frac{1}{799006685782884121} a^{29}$, $\frac{1}{15181127029874798299} a^{30}$, $\frac{1}{15181127029874798299} a^{31}$

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

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

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

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
19Data not computed