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Properties

Label 16.0.28774208330...3744.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 7^{4}\cdot 17^{8}$
Root discriminant $29.25$
Ramified primes $2, 7, 17$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_2\times D_4:D_4$ (as 16T265)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104, -368, -184, 2368, -1220, -2128, 4320, -3616, 1454, 404, -782, 384, -25, -60, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 60*x^13 - 25*x^12 + 384*x^11 - 782*x^10 + 404*x^9 + 1454*x^8 - 3616*x^7 + 4320*x^6 - 2128*x^5 - 1220*x^4 + 2368*x^3 - 184*x^2 - 368*x + 104)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 60*x^13 - 25*x^12 + 384*x^11 - 782*x^10 + 404*x^9 + 1454*x^8 - 3616*x^7 + 4320*x^6 - 2128*x^5 - 1220*x^4 + 2368*x^3 - 184*x^2 - 368*x + 104, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 60 x^{13} - 25 x^{12} + 384 x^{11} - 782 x^{10} + 404 x^{9} + 1454 x^{8} - 3616 x^{7} + 4320 x^{6} - 2128 x^{5} - 1220 x^{4} + 2368 x^{3} - 184 x^{2} - 368 x + 104 \)

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:  \(287742083309962730143744=2^{34}\cdot 7^{4}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.25$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{8} + \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{8} + \frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{1312} a^{14} + \frac{15}{1312} a^{13} + \frac{1}{41} a^{12} - \frac{15}{656} a^{11} - \frac{29}{1312} a^{10} + \frac{63}{1312} a^{9} - \frac{19}{164} a^{8} + \frac{15}{328} a^{7} + \frac{11}{164} a^{6} + \frac{85}{328} a^{5} - \frac{43}{164} a^{4} + \frac{13}{41} a^{3} - \frac{59}{328} a^{2} - \frac{73}{328} a + \frac{2}{41}$, $\frac{1}{7312742418658144} a^{15} - \frac{106974969137}{3656371209329072} a^{14} - \frac{17092296324689}{7312742418658144} a^{13} + \frac{66745645757169}{3656371209329072} a^{12} - \frac{114786304538485}{7312742418658144} a^{11} - \frac{18071268701103}{3656371209329072} a^{10} - \frac{232271986360963}{7312742418658144} a^{9} + \frac{446156639246681}{3656371209329072} a^{8} + \frac{218926547471215}{1828185604664536} a^{7} + \frac{208492608930507}{1828185604664536} a^{6} + \frac{515905217685513}{1828185604664536} a^{5} - \frac{58625083977750}{228523200583067} a^{4} - \frac{15478855126503}{96220294982344} a^{3} - \frac{71634689736127}{457046401166134} a^{2} + \frac{246968816541929}{1828185604664536} a - \frac{30105311969931}{70314830948636}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( \frac{688386944971}{89179785593392} a^{15} - \frac{5447779220025}{89179785593392} a^{14} + \frac{10815883567867}{44589892796696} a^{13} - \frac{2507378648197}{5573736599587} a^{12} - \frac{17635421615445}{89179785593392} a^{11} + \frac{255858035471339}{89179785593392} a^{10} - \frac{32125532444367}{5573736599587} a^{9} + \frac{133081688642957}{44589892796696} a^{8} + \frac{234007567635597}{22294946398348} a^{7} - \frac{582545102163125}{22294946398348} a^{6} + \frac{179357147760931}{5573736599587} a^{5} - \frac{199019863592921}{11147473199174} a^{4} - \frac{5939501373711}{1173418231492} a^{3} + \frac{288973287660795}{22294946398348} a^{2} + \frac{2994766627261}{5573736599587} a - \frac{902969738147}{857497938398} \) (order $4$)
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:  \( 325927.116113 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_4:D_4$ (as 16T265):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2\times D_4:D_4$
Character table for $C_2\times D_4:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-17}) \), 4.4.9248.1, 4.0.2312.1, \(\Q(i, \sqrt{17})\), 8.8.536415961088.2, 8.0.134103990272.4, 8.0.342102016.5

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.9.1$x^{4} + 6 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.9.1$x^{4} + 6 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$