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Properties

Label 16.0.28179280429...000.18
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{8}\cdot 5^{8}$
Root discriminant $21.91$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2 \times (C_4\times C_2):C_2$ (as 16T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![481, 2052, 1088, -300, 6844, -7456, 10940, -8520, 7335, -4040, 2420, -928, 404, -100, 32, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 32*x^14 - 100*x^13 + 404*x^12 - 928*x^11 + 2420*x^10 - 4040*x^9 + 7335*x^8 - 8520*x^7 + 10940*x^6 - 7456*x^5 + 6844*x^4 - 300*x^3 + 1088*x^2 + 2052*x + 481)
 
gp: K = bnfinit(x^16 - 4*x^15 + 32*x^14 - 100*x^13 + 404*x^12 - 928*x^11 + 2420*x^10 - 4040*x^9 + 7335*x^8 - 8520*x^7 + 10940*x^6 - 7456*x^5 + 6844*x^4 - 300*x^3 + 1088*x^2 + 2052*x + 481, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 32 x^{14} - 100 x^{13} + 404 x^{12} - 928 x^{11} + 2420 x^{10} - 4040 x^{9} + 7335 x^{8} - 8520 x^{7} + 10940 x^{6} - 7456 x^{5} + 6844 x^{4} - 300 x^{3} + 1088 x^{2} + 2052 x + 481 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2817928042905600000000=2^{40}\cdot 3^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.91$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{453504790397956820708159} a^{15} + \frac{84362855579352310984135}{453504790397956820708159} a^{14} + \frac{216783314796593521496310}{453504790397956820708159} a^{13} - \frac{44835795857385482248345}{453504790397956820708159} a^{12} - \frac{226133466891318454767694}{453504790397956820708159} a^{11} - \frac{14965898724259142359616}{34884983876765909285243} a^{10} + \frac{152590898376892990056707}{453504790397956820708159} a^{9} - \frac{63795338975763515483271}{453504790397956820708159} a^{8} + \frac{33864947995475581086689}{453504790397956820708159} a^{7} + \frac{101106983369259736933018}{453504790397956820708159} a^{6} + \frac{147941876267049950280326}{453504790397956820708159} a^{5} + \frac{207272131926589931214655}{453504790397956820708159} a^{4} + \frac{150458941842472035931510}{453504790397956820708159} a^{3} - \frac{128504237494766956530008}{453504790397956820708159} a^{2} + \frac{5236640058405095899882}{453504790397956820708159} a + \frac{3118365913838100522634}{34884983876765909285243}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}$, which has order $4$ (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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{6390320664779526378}{1881762615759156932399} a^{15} + \frac{25242093090669797052}{1881762615759156932399} a^{14} - \frac{193355886477336250238}{1881762615759156932399} a^{13} + \frac{594936306216511661984}{1881762615759156932399} a^{12} - \frac{2277757385644700292090}{1881762615759156932399} a^{11} + \frac{388746998065888538268}{144750970443012071723} a^{10} - \frac{12280366141692585662350}{1881762615759156932399} a^{9} + \frac{19398342970838963432476}{1881762615759156932399} a^{8} - \frac{32081775881453278623878}{1881762615759156932399} a^{7} + \frac{34096998689172835760250}{1881762615759156932399} a^{6} - \frac{38778629267380595871246}{1881762615759156932399} a^{5} + \frac{22207309604970974402273}{1881762615759156932399} a^{4} - \frac{18660116159885336866412}{1881762615759156932399} a^{3} - \frac{1019264956854439670322}{1881762615759156932399} a^{2} - \frac{2873012245665627956864}{1881762615759156932399} a - \frac{236664777441252900804}{144750970443012071723} \) (order $12$)
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:  \( 25454.3718839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:C_2$ (as 16T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2 \times (C_4\times C_2):C_2$
Character table for $C_2 \times (C_4\times C_2):C_2$

Intermediate fields

\(\Q(\sqrt{30}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(i, \sqrt{30})\), 8.0.3317760000.2, 8.0.5308416.2, 8.0.3317760000.16

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

Sibling fields

Galois closure: data not computed
Degree 16 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$