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Properties

Label 16.0.41937406987...7921.4
Degree $16$
Signature $[0, 8]$
Discriminant $11^{8}\cdot 89^{14}$
Root discriminant $168.43$
Ramified primes $11, 89$
Class number $32$ (GRH)
Class group $[2, 2, 8]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2688284, -5449640, 40870017, 54084660, 41196780, 26533814, 16017095, 7653230, 2751785, 735470, 145223, 16650, -207, -454, -54, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 54*x^14 - 454*x^13 - 207*x^12 + 16650*x^11 + 145223*x^10 + 735470*x^9 + 2751785*x^8 + 7653230*x^7 + 16017095*x^6 + 26533814*x^5 + 41196780*x^4 + 54084660*x^3 + 40870017*x^2 - 5449640*x + 2688284)
 
gp: K = bnfinit(x^16 - 2*x^15 - 54*x^14 - 454*x^13 - 207*x^12 + 16650*x^11 + 145223*x^10 + 735470*x^9 + 2751785*x^8 + 7653230*x^7 + 16017095*x^6 + 26533814*x^5 + 41196780*x^4 + 54084660*x^3 + 40870017*x^2 - 5449640*x + 2688284, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 54 x^{14} - 454 x^{13} - 207 x^{12} + 16650 x^{11} + 145223 x^{10} + 735470 x^{9} + 2751785 x^{8} + 7653230 x^{7} + 16017095 x^{6} + 26533814 x^{5} + 41196780 x^{4} + 54084660 x^{3} + 40870017 x^{2} - 5449640 x + 2688284 \)

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:  \(419374069872350970710519002168327921=11^{8}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.43$
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:  $11, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{5}{32} a^{4} + \frac{1}{32} a^{3} + \frac{15}{32} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{32} a^{11} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{7}{32} a^{5} + \frac{1}{8} a^{4} - \frac{1}{32} a^{2} - \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{704} a^{12} + \frac{5}{352} a^{11} - \frac{1}{64} a^{10} - \frac{1}{22} a^{9} + \frac{27}{704} a^{8} - \frac{1}{44} a^{7} - \frac{11}{64} a^{6} - \frac{37}{352} a^{5} - \frac{17}{704} a^{4} - \frac{7}{88} a^{3} + \frac{193}{704} a^{2} + \frac{37}{88} a - \frac{85}{176}$, $\frac{1}{704} a^{13} - \frac{1}{704} a^{11} - \frac{5}{352} a^{10} - \frac{5}{704} a^{9} - \frac{49}{704} a^{7} - \frac{1}{88} a^{6} + \frac{85}{704} a^{5} + \frac{57}{352} a^{4} - \frac{127}{704} a^{3} - \frac{5}{22} a^{2} - \frac{3}{16} a - \frac{1}{22}$, $\frac{1}{5778640384} a^{14} + \frac{2201065}{5778640384} a^{13} - \frac{461627}{722330048} a^{12} - \frac{37806715}{5778640384} a^{11} + \frac{36242}{11286407} a^{10} + \frac{76593553}{5778640384} a^{9} - \frac{31886831}{2889320192} a^{8} - \frac{712041425}{5778640384} a^{7} - \frac{333880871}{1444660096} a^{6} + \frac{50808557}{525330944} a^{5} - \frac{16166449}{722330048} a^{4} + \frac{291091539}{5778640384} a^{3} + \frac{1396190469}{5778640384} a^{2} - \frac{108290125}{1444660096} a - \frac{76819429}{1444660096}$, $\frac{1}{2955149859972867726412787004082688} a^{15} + \frac{20830239883278906412051}{268649987270260702401162454916608} a^{14} - \frac{95744932818186734757999660411}{369393732496608465801598375510336} a^{13} - \frac{358852718065895680575235166571}{2955149859972867726412787004082688} a^{12} - \frac{87136762506100674823890657531}{8395312102195646950036326716144} a^{11} + \frac{33649754674223008506883530737009}{2955149859972867726412787004082688} a^{10} + \frac{70553932548293841612069736454569}{1477574929986433863206393502041344} a^{9} + \frac{162002568396718177957673940115903}{2955149859972867726412787004082688} a^{8} - \frac{3681153826285430198633005454869}{67162496817565175600290613729152} a^{7} - \frac{61600418006859850430339014569507}{268649987270260702401162454916608} a^{6} - \frac{56258869252996027042991961020365}{369393732496608465801598375510336} a^{5} + \frac{242817331768532366677036992857715}{2955149859972867726412787004082688} a^{4} - \frac{1168840772151270878981872148259819}{2955149859972867726412787004082688} a^{3} + \frac{204212378749021814209175218778933}{738787464993216931603196751020672} a^{2} - \frac{6340792247069420857529361223663}{67162496817565175600290613729152} a - \frac{601725881669991948646350170801}{92348433124152116450399593877584}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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:  \( -1 \) (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:  \( 19563150221.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-979}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-11}, \sqrt{89})\), 4.4.704969.1, 4.0.85301249.2, 8.0.7276303080960001.2, 8.4.5351991522359009.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$89$89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.3$x^{8} - 7209$$8$$1$$7$$C_8$$[\ ]_{8}$