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Properties

Label 16.8.97452031169...5489.1
Degree $16$
Signature $[8, 4]$
Discriminant $41^{8}\cdot 73^{14}$
Root discriminant $273.40$
Ramified primes $41, 73$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5383323648, 21852549120, -28941778432, 12970627968, -1437895264, 1035976504, -98745148, -46338526, 8653049, 41572, -50700, -20048, 2366, 788, -144, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 144*x^14 + 788*x^13 + 2366*x^12 - 20048*x^11 - 50700*x^10 + 41572*x^9 + 8653049*x^8 - 46338526*x^7 - 98745148*x^6 + 1035976504*x^5 - 1437895264*x^4 + 12970627968*x^3 - 28941778432*x^2 + 21852549120*x - 5383323648)
 
gp: K = bnfinit(x^16 - 2*x^15 - 144*x^14 + 788*x^13 + 2366*x^12 - 20048*x^11 - 50700*x^10 + 41572*x^9 + 8653049*x^8 - 46338526*x^7 - 98745148*x^6 + 1035976504*x^5 - 1437895264*x^4 + 12970627968*x^3 - 28941778432*x^2 + 21852549120*x - 5383323648, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 144 x^{14} + 788 x^{13} + 2366 x^{12} - 20048 x^{11} - 50700 x^{10} + 41572 x^{9} + 8653049 x^{8} - 46338526 x^{7} - 98745148 x^{6} + 1035976504 x^{5} - 1437895264 x^{4} + 12970627968 x^{3} - 28941778432 x^{2} + 21852549120 x - 5383323648 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(974520311694399191366085457215090805489=41^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $273.40$
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:  $41, 73$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{48} a^{6} - \frac{1}{16} a^{5} - \frac{1}{48} a^{4} - \frac{1}{48} a^{3} - \frac{1}{6} a^{2} - \frac{1}{12} a$, $\frac{1}{96} a^{7} - \frac{1}{24} a^{5} + \frac{1}{48} a^{4} + \frac{19}{96} a^{3} + \frac{7}{48} a^{2}$, $\frac{1}{384} a^{8} + \frac{1}{192} a^{6} + \frac{1}{48} a^{5} - \frac{23}{384} a^{4} - \frac{11}{48} a^{3} + \frac{7}{32} a^{2} + \frac{3}{8} a$, $\frac{1}{1152} a^{9} + \frac{1}{1152} a^{8} - \frac{1}{192} a^{7} + \frac{1}{576} a^{6} - \frac{7}{1152} a^{5} + \frac{25}{1152} a^{4} - \frac{49}{288} a^{3} + \frac{19}{96} a^{2} - \frac{19}{72} a + \frac{1}{3}$, $\frac{1}{4608} a^{10} + \frac{1}{4608} a^{9} - \frac{1}{768} a^{8} - \frac{11}{2304} a^{7} - \frac{31}{4608} a^{6} - \frac{239}{4608} a^{5} - \frac{19}{1152} a^{4} + \frac{43}{384} a^{3} + \frac{71}{288} a^{2} + \frac{1}{6} a$, $\frac{1}{9216} a^{11} - \frac{1}{9216} a^{10} - \frac{1}{4608} a^{8} - \frac{35}{9216} a^{7} + \frac{31}{9216} a^{6} + \frac{173}{4608} a^{5} - \frac{119}{2304} a^{4} + \frac{1}{128} a^{3} + \frac{35}{144} a^{2} + \frac{35}{72} a + \frac{1}{3}$, $\frac{1}{2267136} a^{12} - \frac{23}{566784} a^{11} - \frac{11}{2267136} a^{10} + \frac{1}{47232} a^{9} + \frac{2543}{2267136} a^{8} + \frac{419}{188928} a^{7} - \frac{8801}{2267136} a^{6} - \frac{553}{283392} a^{5} + \frac{8431}{566784} a^{4} - \frac{7697}{47232} a^{3} - \frac{6877}{35424} a^{2} - \frac{155}{984} a - \frac{25}{123}$, $\frac{1}{36274176} a^{13} - \frac{1}{9068544} a^{12} + \frac{995}{36274176} a^{11} - \frac{1321}{18137088} a^{10} + \frac{371}{36274176} a^{9} - \frac{4547}{4534272} a^{8} - \frac{35459}{36274176} a^{7} - \frac{163997}{18137088} a^{6} + \frac{222539}{9068544} a^{5} + \frac{59717}{4534272} a^{4} + \frac{5581}{70848} a^{3} + \frac{51691}{283392} a^{2} + \frac{1255}{3936} a - \frac{29}{246}$, $\frac{1}{116222459904} a^{14} + \frac{937}{116222459904} a^{13} - \frac{2881}{116222459904} a^{12} + \frac{2640373}{116222459904} a^{11} - \frac{1401517}{38740819968} a^{10} - \frac{2560907}{38740819968} a^{9} - \frac{631505}{4304535552} a^{8} - \frac{572423393}{116222459904} a^{7} - \frac{410901947}{58111229952} a^{6} - \frac{1222996391}{29055614976} a^{5} - \frac{168776797}{4842602496} a^{4} + \frac{21497947}{302662656} a^{3} + \frac{26553583}{907987968} a^{2} - \frac{11550563}{37832832} a - \frac{263755}{788184}$, $\frac{1}{4100378375515428037447925823700992} a^{15} - \frac{3045984138589674457231}{4100378375515428037447925823700992} a^{14} + \frac{716747141538426711210829}{151865865759830668053626882359296} a^{13} - \frac{164297702369361580856226961}{1366792791838476012482641941233664} a^{12} + \frac{62548448987604126270596105065}{4100378375515428037447925823700992} a^{11} + \frac{30568526205038728662523010623}{455597597279492004160880647077888} a^{10} - \frac{78822038110010055266392276505}{1366792791838476012482641941233664} a^{9} + \frac{3615319747284252897627508054471}{4100378375515428037447925823700992} a^{8} - \frac{1121965501843734676870149824203}{227798798639746002080440323538944} a^{7} + \frac{1250302962257248071911764188923}{341698197959619003120660485308416} a^{6} - \frac{9784521755392854643144209853423}{512547296939428504680990727962624} a^{5} - \frac{157956343976810549678952390503}{3559356228746031282506880055296} a^{4} - \frac{4351661067982436589795086851385}{32034206058714281542561920497664} a^{3} + \frac{411465659551579034780033861039}{4004275757339285192820240062208} a^{2} - \frac{18699239310236619147616827545}{83422411611235108183755001296} a - \frac{71234599481997735827148682}{1737966908567398087161562527}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $11$
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:  \( 607131923951000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{73}) \), \(\Q(\sqrt{2993}) \), \(\Q(\sqrt{41}) \), 4.4.389017.1, 4.4.653937577.1, \(\Q(\sqrt{41}, \sqrt{73})\), 8.4.18570676910602057.1 x2, 8.4.31217307886722057817.1 x2, 8.8.427634354612630929.1

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 8 siblings: 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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
41Data not computed
$73$73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.3$x^{8} - 45625$$8$$1$$7$$C_8$$[\ ]_{8}$