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Properties

Label 16.8.14014641336...5089.2
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{15}$
Root discriminant $765.88$
Ramified primes $37, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-225600707201264, -127935155601144, -10440602463500, 7302425346538, 1753792080987, -98834863124, -50306428611, -858373597, 314500669, 41129851, 1863479, -123760, -29059, -2156, 94, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 94*x^14 - 2156*x^13 - 29059*x^12 - 123760*x^11 + 1863479*x^10 + 41129851*x^9 + 314500669*x^8 - 858373597*x^7 - 50306428611*x^6 - 98834863124*x^5 + 1753792080987*x^4 + 7302425346538*x^3 - 10440602463500*x^2 - 127935155601144*x - 225600707201264)
 
gp: K = bnfinit(x^16 - x^15 + 94*x^14 - 2156*x^13 - 29059*x^12 - 123760*x^11 + 1863479*x^10 + 41129851*x^9 + 314500669*x^8 - 858373597*x^7 - 50306428611*x^6 - 98834863124*x^5 + 1753792080987*x^4 + 7302425346538*x^3 - 10440602463500*x^2 - 127935155601144*x - 225600707201264, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 94 x^{14} - 2156 x^{13} - 29059 x^{12} - 123760 x^{11} + 1863479 x^{10} + 41129851 x^{9} + 314500669 x^{8} - 858373597 x^{7} - 50306428611 x^{6} - 98834863124 x^{5} + 1753792080987 x^{4} + 7302425346538 x^{3} - 10440602463500 x^{2} - 127935155601144 x - 225600707201264 \)

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:  \(14014641336002292789493279203934745080101945089=37^{14}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $765.88$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8} + \frac{20}{41} a^{7} + \frac{11}{41} a^{6} + \frac{14}{41} a^{5} + \frac{13}{41} a^{4} - \frac{15}{41} a^{3} + \frac{12}{41} a^{2} - \frac{9}{41} a + \frac{10}{41}$, $\frac{1}{82} a^{9} - \frac{1}{82} a^{8} - \frac{20}{41} a^{7} + \frac{29}{82} a^{6} - \frac{35}{82} a^{5} - \frac{1}{82} a^{4} + \frac{20}{41} a^{3} - \frac{15}{82} a^{2} + \frac{35}{82} a + \frac{18}{41}$, $\frac{1}{82} a^{10} - \frac{1}{82} a^{8} - \frac{31}{82} a^{7} + \frac{12}{41} a^{6} + \frac{16}{41} a^{5} - \frac{15}{82} a^{4} - \frac{1}{82} a^{3} + \frac{4}{41} a^{2} + \frac{39}{82} a + \frac{13}{41}$, $\frac{1}{82} a^{11} - \frac{16}{41} a^{7} + \frac{3}{82} a^{6} - \frac{6}{41} a^{5} + \frac{2}{41} a^{4} - \frac{11}{41} a^{3} - \frac{1}{41} a^{2} + \frac{19}{82} a + \frac{14}{41}$, $\frac{1}{82} a^{12} - \frac{13}{82} a^{7} + \frac{6}{41} a^{6} - \frac{20}{41} a^{5} - \frac{8}{41} a^{4} + \frac{5}{41} a^{3} - \frac{7}{82} a^{2} - \frac{7}{41} a - \frac{4}{41}$, $\frac{1}{82} a^{13} - \frac{1}{82} a^{8} + \frac{3}{41} a^{7} + \frac{5}{41} a^{6} - \frac{6}{41} a^{5} + \frac{1}{41} a^{4} - \frac{23}{82} a^{3} - \frac{17}{41} a^{2} - \frac{17}{41} a + \frac{19}{41}$, $\frac{1}{7052} a^{14} - \frac{3}{7052} a^{13} - \frac{15}{3526} a^{12} + \frac{7}{3526} a^{11} + \frac{11}{7052} a^{10} - \frac{21}{3526} a^{9} + \frac{61}{7052} a^{8} - \frac{3083}{7052} a^{7} + \frac{2237}{7052} a^{6} + \frac{2837}{7052} a^{5} + \frac{3293}{7052} a^{4} + \frac{612}{1763} a^{3} - \frac{1243}{7052} a^{2} + \frac{787}{1763} a + \frac{5}{41}$, $\frac{1}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{15} + \frac{21844370486067206970981643063939585513155828188265340464909601657955506180790215}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{14} - \frac{1036435301269614679867771322787195845852068354368819381555796891868703819956538607}{220230195107598179153499164788322627194222257169357707685905759682388497991764121212} a^{13} - \frac{1646308816916768733578568592006897553024023629238693216546443266737393772698263}{283072230215421824104754710524836281740645574767811963606562673113609894590956454} a^{12} - \frac{196343435893325293557838763371460672950168286722031416862425897816602057454410083}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{11} + \frac{377502152899763907593707151654523572309179368649450885006571443602865198328305589}{110115097553799089576749582394161313597111128584678853842952879841194248995882060606} a^{10} - \frac{1616634044479600963696525619081347034599886408779138179675239314570345269969754221}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{9} + \frac{3619872604026229571118404330975521451351920242660302826924986062799624786873895743}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{8} + \frac{178826099178526177202849574174113403560569864226711271419903168446876891557255402057}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{7} + \frac{4350053939385956567019687657026984208333312469197097098122642254871894275105676501}{10243264888725496704813914641317331497405686379970125938879337659645976650779726568} a^{6} + \frac{116339519112246716874315653633483210517712939053777479577947466692328061141622061921}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{5} + \frac{9972139850310777990461261293662000088842044614311040076774448453009840661945082957}{110115097553799089576749582394161313597111128584678853842952879841194248995882060606} a^{4} - \frac{8887907685032596170788236548990922002484996482516292481540261275159881705773974645}{440460390215196358306998329576645254388444514338715415371811519364776995983528242424} a^{3} - \frac{12438420598798982403062423400565646093459885263262781053501545250357595912983133909}{220230195107598179153499164788322627194222257169357707685905759682388497991764121212} a^{2} - \frac{25111481137782635566621882118983752076910383783152249729436879999309561443835596279}{110115097553799089576749582394161313597111128584678853842952879841194248995882060606} a + \frac{125502600626237615462678387124565942667804462687518530000325395364756365392880563}{1280408111090687088101739330164666437175710797496265742359917207455747081347465821}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 15178902125900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.94352849.1, 8.8.499686183762100623329.1

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{12}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
41Data not computed