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Properties

Label 16.0.24968628224...2241.4
Degree $16$
Signature $[0, 8]$
Discriminant $37^{12}\cdot 41^{14}$
Root discriminant $386.67$
Ramified primes $37, 41$
Class number $4672$ (GRH)
Class group $[2, 2, 2, 584]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5489864782943, -2040068815012, 1231014931084, -313193327620, 126145294036, -23152161472, 6658827530, -877574404, 217796337, -13294364, 3755888, -86770, 30169, -468, 153, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 153*x^14 - 468*x^13 + 30169*x^12 - 86770*x^11 + 3755888*x^10 - 13294364*x^9 + 217796337*x^8 - 877574404*x^7 + 6658827530*x^6 - 23152161472*x^5 + 126145294036*x^4 - 313193327620*x^3 + 1231014931084*x^2 - 2040068815012*x + 5489864782943)
 
gp: K = bnfinit(x^16 - 2*x^15 + 153*x^14 - 468*x^13 + 30169*x^12 - 86770*x^11 + 3755888*x^10 - 13294364*x^9 + 217796337*x^8 - 877574404*x^7 + 6658827530*x^6 - 23152161472*x^5 + 126145294036*x^4 - 313193327620*x^3 + 1231014931084*x^2 - 2040068815012*x + 5489864782943, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 153 x^{14} - 468 x^{13} + 30169 x^{12} - 86770 x^{11} + 3755888 x^{10} - 13294364 x^{9} + 217796337 x^{8} - 877574404 x^{7} + 6658827530 x^{6} - 23152161472 x^{5} + 126145294036 x^{4} - 313193327620 x^{3} + 1231014931084 x^{2} - 2040068815012 x + 5489864782943 \)

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:  \(249686282242731792647174886492450339042241=37^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $386.67$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} + \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{3}{10} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{3}{10}$, $\frac{1}{10} a^{13} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{10} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{10} a^{3} + \frac{3}{10} a^{2} - \frac{1}{10} a - \frac{3}{10}$, $\frac{1}{7130} a^{14} + \frac{45}{1426} a^{13} + \frac{14}{713} a^{12} + \frac{776}{3565} a^{11} + \frac{801}{7130} a^{10} - \frac{661}{3565} a^{9} + \frac{317}{7130} a^{8} + \frac{2443}{7130} a^{7} + \frac{982}{3565} a^{6} + \frac{1699}{7130} a^{5} - \frac{136}{713} a^{4} - \frac{249}{1426} a^{3} + \frac{214}{3565} a^{2} - \frac{405}{1426} a - \frac{17}{230}$, $\frac{1}{586638792864545394841992441757874334100335727720841769483736885962264193102394086570} a^{15} + \frac{3017070768760972180581988124522247770187726103552145112700195892139172181636679}{117327758572909078968398488351574866820067145544168353896747377192452838620478817314} a^{14} - \frac{9020952377962587299293327779583344746267255473000084021870707615932622939815550939}{586638792864545394841992441757874334100335727720841769483736885962264193102394086570} a^{13} + \frac{4544794416644682372289872200603918601162103053986637691471763547246812168633964801}{293319396432272697420996220878937167050167863860420884741868442981132096551197043285} a^{12} - \frac{12646911695668867131979791558843289178610419942454401286799295961815566578006765569}{293319396432272697420996220878937167050167863860420884741868442981132096551197043285} a^{11} + \frac{20325981327941947170365848344812534387719258591749411531694360170184882141773849782}{293319396432272697420996220878937167050167863860420884741868442981132096551197043285} a^{10} - \frac{6011338785700600259038797197532449644101176024048574664591194083025176345672549326}{58663879286454539484199244175787433410033572772084176948373688596226419310239408657} a^{9} + \frac{17144666105939034636591851656752708751062917519235337458817888513838760839412131515}{117327758572909078968398488351574866820067145544168353896747377192452838620478817314} a^{8} - \frac{5136835571293978888368136716200025578511646665463529858188714850710088712173042291}{58663879286454539484199244175787433410033572772084176948373688596226419310239408657} a^{7} - \frac{5711367873408539233873240023365716328634943312487048428779368920921872416627147939}{117327758572909078968398488351574866820067145544168353896747377192452838620478817314} a^{6} + \frac{51423958295739553308120479902711948870817653177456659466142874602608829788071418681}{293319396432272697420996220878937167050167863860420884741868442981132096551197043285} a^{5} + \frac{19026638499607156265801394797796176236805282740178700572667220723002969475791551116}{293319396432272697420996220878937167050167863860420884741868442981132096551197043285} a^{4} - \frac{157202306687762616142214726658697787137997163540589761946101944541260745159642801913}{586638792864545394841992441757874334100335727720841769483736885962264193102394086570} a^{3} - \frac{29734878323383724563528973408779458531836757692689783659567657748782103461133993127}{586638792864545394841992441757874334100335727720841769483736885962264193102394086570} a^{2} + \frac{4082211697473763328132913703181823049966264250511097453530694605425645547795806956}{293319396432272697420996220878937167050167863860420884741868442981132096551197043285} a + \frac{1359147452922190994109870418316175904238997857844624076609765363043451759688382026}{9461916013944280561967620028352811840327995608400673701350594934875228921006356235}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{584}$, which has order $4672$ (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:  \( 36382443651.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.0.365000864691088841.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
37Data not computed
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$