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Properties

Label 16.0.36612584972...1936.7
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 7^{8}\cdot 41^{12}$
Root discriminant $342.95$
Ramified primes $2, 7, 41$
Class number $123863040$ (GRH)
Class group $[2, 4, 4, 8, 24, 24, 840]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1694656660932, -291056903424, 499235025024, -43796710368, 33449145528, -3101634256, 314280384, -63183296, -2998808, -556944, 294176, -16400, 4356, -328, 32, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 32*x^14 - 328*x^13 + 4356*x^12 - 16400*x^11 + 294176*x^10 - 556944*x^9 - 2998808*x^8 - 63183296*x^7 + 314280384*x^6 - 3101634256*x^5 + 33449145528*x^4 - 43796710368*x^3 + 499235025024*x^2 - 291056903424*x + 1694656660932)
 
gp: K = bnfinit(x^16 + 32*x^14 - 328*x^13 + 4356*x^12 - 16400*x^11 + 294176*x^10 - 556944*x^9 - 2998808*x^8 - 63183296*x^7 + 314280384*x^6 - 3101634256*x^5 + 33449145528*x^4 - 43796710368*x^3 + 499235025024*x^2 - 291056903424*x + 1694656660932, 1)
 

Normalized defining polynomial

\( x^{16} + 32 x^{14} - 328 x^{13} + 4356 x^{12} - 16400 x^{11} + 294176 x^{10} - 556944 x^{9} - 2998808 x^{8} - 63183296 x^{7} + 314280384 x^{6} - 3101634256 x^{5} + 33449145528 x^{4} - 43796710368 x^{3} + 499235025024 x^{2} - 291056903424 x + 1694656660932 \)

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:  \(36612584972217083233681138272216794791936=2^{48}\cdot 7^{8}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $342.95$
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:  $2, 7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4592=2^{4}\cdot 7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4592}(1,·)$, $\chi_{4592}(4355,·)$, $\chi_{4592}(2715,·)$, $\chi_{4592}(3781,·)$, $\chi_{4592}(2953,·)$, $\chi_{4592}(2295,·)$, $\chi_{4592}(4173,·)$, $\chi_{4592}(1231,·)$, $\chi_{4592}(155,·)$, $\chi_{4592}(2141,·)$, $\chi_{4592}(3935,·)$, $\chi_{4592}(3107,·)$, $\chi_{4592}(2533,·)$, $\chi_{4592}(1065,·)$, $\chi_{4592}(4017,·)$, $\chi_{4592}(2871,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{72} a^{8} - \frac{1}{12} a^{4} - \frac{1}{18} a^{2} + \frac{1}{4}$, $\frac{1}{648} a^{9} + \frac{1}{27} a^{7} - \frac{13}{108} a^{5} + \frac{11}{162} a^{3} + \frac{5}{36} a$, $\frac{1}{7128} a^{10} + \frac{1}{297} a^{8} - \frac{4}{99} a^{7} - \frac{49}{1188} a^{6} + \frac{14}{99} a^{5} + \frac{281}{1782} a^{4} + \frac{5}{99} a^{3} + \frac{29}{396} a^{2} + \frac{16}{33} a - \frac{4}{11}$, $\frac{1}{7128} a^{11} + \frac{1}{3564} a^{9} + \frac{1}{792} a^{8} - \frac{5}{1188} a^{7} + \frac{1}{33} a^{6} - \frac{140}{891} a^{5} + \frac{1}{44} a^{4} + \frac{173}{3564} a^{3} + \frac{41}{198} a^{2} - \frac{61}{198} a - \frac{1}{4}$, $\frac{1}{84103272} a^{12} - \frac{587}{14017212} a^{11} - \frac{3133}{84103272} a^{10} - \frac{12751}{28034424} a^{9} + \frac{2437}{1218888} a^{8} + \frac{4207}{778734} a^{7} - \frac{913655}{42051636} a^{6} - \frac{1642567}{14017212} a^{5} + \frac{37739}{1911438} a^{4} - \frac{11180}{152361} a^{3} + \frac{102647}{424764} a^{2} + \frac{542605}{1557468} a + \frac{6559}{15732}$, $\frac{1}{252309816} a^{13} - \frac{1}{252309816} a^{12} + \frac{8453}{252309816} a^{11} - \frac{13883}{252309816} a^{10} - \frac{194}{318573} a^{9} + \frac{15361}{84103272} a^{8} - \frac{5806109}{126154908} a^{7} + \frac{2441171}{126154908} a^{6} - \frac{17157139}{126154908} a^{5} - \frac{5922979}{63077454} a^{4} + \frac{6480391}{42051636} a^{3} - \frac{465505}{14017212} a^{2} + \frac{923683}{2336202} a - \frac{216089}{519156}$, $\frac{1}{293207762170415664} a^{14} + \frac{52401787}{73301940542603916} a^{13} - \frac{773555}{708231309590376} a^{12} + \frac{1931942632277}{73301940542603916} a^{11} - \frac{967937513233}{18325485135650979} a^{10} - \frac{4677187092511}{12216990090433986} a^{9} - \frac{424469555467}{1409652702742383} a^{8} - \frac{243282175686520}{18325485135650979} a^{7} + \frac{2238827140827857}{48867960361735944} a^{6} + \frac{2779366346860043}{36650970271301958} a^{5} + \frac{9337496072268961}{73301940542603916} a^{4} + \frac{1634400503993}{29870391419154} a^{3} - \frac{467063815609525}{8144660060289324} a^{2} - \frac{222850915390672}{678721671690777} a - \frac{18724981647512}{75413519076753}$, $\frac{1}{1041471973390582093196599996033377081648} a^{15} + \frac{294874868462561165}{183325466183872926103960569623900208} a^{14} - \frac{63876050799341789236181374987}{173578662231763682199433332672229513608} a^{13} + \frac{447396398339301761570977105777}{520735986695291046598299998016688540824} a^{12} - \frac{7732819974296259547087513569428945}{520735986695291046598299998016688540824} a^{11} - \frac{169661182017434472448243620758867}{28929777038627280366572222112038252268} a^{10} - \frac{7557082777472613037256335603979069}{27407157194489002452542105158773081096} a^{9} + \frac{461699647334324328829721341870877387}{260367993347645523299149999008344270412} a^{8} - \frac{773743005079909539868537730752446233}{19286518025751520244381481408025501512} a^{7} + \frac{4196929521836454936793328343627729755}{520735986695291046598299998016688540824} a^{6} + \frac{8581558851011814761675575297429381585}{65091998336911380824787499752086067603} a^{5} + \frac{140106947457206104630904942807750927}{86789331115881841099716666336114756804} a^{4} + \frac{89445868083555011074481262910568890}{2410814753218940030547685176003187689} a^{3} + \frac{267133481044209853804022924480471995}{9643259012875760122190740704012750756} a^{2} + \frac{683766868230234978080926667678754653}{3214419670958586707396913568004250252} a + \frac{72752613795564731391815809775565179}{178578870608810372633161864889125014}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{24}\times C_{24}\times C_{840}$, which has order $123863040$ (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:  \( 411276342.2126948 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-287}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{574}) \), \(\Q(\sqrt{82}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-41}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-2}, \sqrt{-287})\), \(\Q(\sqrt{-14}, \sqrt{82})\), \(\Q(\sqrt{7}, \sqrt{-41})\), \(\Q(\sqrt{-2}, \sqrt{-41})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{7}, \sqrt{82})\), \(\Q(\sqrt{-14}, \sqrt{-41})\), 4.4.141150208.1, 4.0.6916360192.4, 4.0.141150208.4, 4.4.6916360192.1, 8.0.444638964023296.21, 8.0.47836038305482276864.4, 8.0.47836038305482276864.5, 8.0.79693524873773056.31, 8.0.191344153221929107456.6, 8.8.191344153221929107456.1, 8.0.191344153221929107456.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.5$x^{8} - 15$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.5$x^{8} - 15$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$