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Properties

Label 16.0.91986798940...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 5^{10}\cdot 41^{2}\cdot 601^{4}$
Root discriminant $86.15$
Ramified primes $2, 5, 41, 601$
Class number $100$ (GRH)
Class group $[10, 10]$ (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11828879, -10679304, 12924496, -7062696, 5119981, -2027400, 1094032, -295880, 136532, -23240, 12094, -1208, 895, -32, 46, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 46*x^14 - 32*x^13 + 895*x^12 - 1208*x^11 + 12094*x^10 - 23240*x^9 + 136532*x^8 - 295880*x^7 + 1094032*x^6 - 2027400*x^5 + 5119981*x^4 - 7062696*x^3 + 12924496*x^2 - 10679304*x + 11828879)
 
gp: K = bnfinit(x^16 + 46*x^14 - 32*x^13 + 895*x^12 - 1208*x^11 + 12094*x^10 - 23240*x^9 + 136532*x^8 - 295880*x^7 + 1094032*x^6 - 2027400*x^5 + 5119981*x^4 - 7062696*x^3 + 12924496*x^2 - 10679304*x + 11828879, 1)
 

Normalized defining polynomial

\( x^{16} + 46 x^{14} - 32 x^{13} + 895 x^{12} - 1208 x^{11} + 12094 x^{10} - 23240 x^{9} + 136532 x^{8} - 295880 x^{7} + 1094032 x^{6} - 2027400 x^{5} + 5119981 x^{4} - 7062696 x^{3} + 12924496 x^{2} - 10679304 x + 11828879 \)

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:  \(9198679894097385226240000000000=2^{32}\cdot 5^{10}\cdot 41^{2}\cdot 601^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.15$
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:  $2, 5, 41, 601$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{52} a^{14} - \frac{5}{26} a^{13} + \frac{9}{52} a^{12} - \frac{3}{13} a^{9} - \frac{3}{26} a^{8} - \frac{2}{13} a^{7} - \frac{1}{26} a^{6} + \frac{6}{13} a^{5} - \frac{7}{26} a^{4} + \frac{3}{52} a^{2} + \frac{9}{26} a - \frac{19}{52}$, $\frac{1}{39838002771182340420543323417437881270158988} a^{15} - \frac{56761478118652240599124532676346786768289}{9959500692795585105135830854359470317539747} a^{14} + \frac{5696061494364688334402721852485914943955221}{39838002771182340420543323417437881270158988} a^{13} + \frac{336298955254827735901497453747839898292094}{3319833564265195035045276951453156772513249} a^{12} + \frac{76992881413771434334346624628623902514260}{766115437907352700395063911873805409041519} a^{11} + \frac{1413232240374828803254120955397172851132083}{9959500692795585105135830854359470317539747} a^{10} + \frac{1143344949216278830367708583995239309946043}{6639667128530390070090553902906313545026498} a^{9} + \frac{222792841753787109986599589804361875854006}{766115437907352700395063911873805409041519} a^{8} - \frac{97848474472601575780858369839632438598725}{510743625271568466930042607915870272694346} a^{7} - \frac{821865459298198809607166641712825760586100}{9959500692795585105135830854359470317539747} a^{6} + \frac{4166624389337311133894016971759549473927135}{19919001385591170210271661708718940635079494} a^{5} - \frac{3596197214312978366358026848202351798006983}{9959500692795585105135830854359470317539747} a^{4} + \frac{5392318391051722677438218708644491548388017}{13279334257060780140181107805812627090052996} a^{3} - \frac{409670088533636010652285410111465742675790}{3319833564265195035045276951453156772513249} a^{2} + \frac{12131866893651463575680649780479343804843301}{39838002771182340420543323417437881270158988} a - \frac{3597289747056469190919576074206894744072367}{9959500692795585105135830854359470317539747}$

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

Class group and class number

$C_{10}\times C_{10}$, which has order $100$ (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:  \( 13690653.5317 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.0.15025.1, 4.0.961600.5, \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.924674560000.4

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.8.16.13$x^{8} + 6 x^{6} + 4 x^{5} + 2 x^{4} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
601Data not computed