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Properties

Label 16.0.19799315994...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{14}\cdot 5^{14}\cdot 7^{14}$
Root discriminant $58.69$
Ramified primes $3, 5, 7$
Class number $144$ (GRH)
Class group $[2, 6, 12]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42331, 80847, 932012, -425649, 629776, -455553, 64505, -60987, 38704, -13866, 5180, -966, 616, -147, 53, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 53*x^14 - 147*x^13 + 616*x^12 - 966*x^11 + 5180*x^10 - 13866*x^9 + 38704*x^8 - 60987*x^7 + 64505*x^6 - 455553*x^5 + 629776*x^4 - 425649*x^3 + 932012*x^2 + 80847*x + 42331)
 
gp: K = bnfinit(x^16 - 6*x^15 + 53*x^14 - 147*x^13 + 616*x^12 - 966*x^11 + 5180*x^10 - 13866*x^9 + 38704*x^8 - 60987*x^7 + 64505*x^6 - 455553*x^5 + 629776*x^4 - 425649*x^3 + 932012*x^2 + 80847*x + 42331, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 53 x^{14} - 147 x^{13} + 616 x^{12} - 966 x^{11} + 5180 x^{10} - 13866 x^{9} + 38704 x^{8} - 60987 x^{7} + 64505 x^{6} - 455553 x^{5} + 629776 x^{4} - 425649 x^{3} + 932012 x^{2} + 80847 x + 42331 \)

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:  \(19799315994393973883056640625=3^{14}\cdot 5^{14}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.69$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{33148933519983126522663205680116929888449877382} a^{15} + \frac{3321092335194133125116015097063533365833513398}{16574466759991563261331602840058464944224938691} a^{14} + \frac{2581087222153971661578709437461075850268312195}{16574466759991563261331602840058464944224938691} a^{13} - \frac{652922249795264995708818044440008670148508407}{33148933519983126522663205680116929888449877382} a^{12} + \frac{7082769943405615659514560811386642424981872314}{16574466759991563261331602840058464944224938691} a^{11} - \frac{1690849408400240020351110590442902076487644614}{16574466759991563261331602840058464944224938691} a^{10} - \frac{9594603083308450868393427304013905007891063367}{33148933519983126522663205680116929888449877382} a^{9} + \frac{1540917119297592476622343673163882511985163592}{16574466759991563261331602840058464944224938691} a^{8} + \frac{6190502958431322771795443708032995954953688037}{16574466759991563261331602840058464944224938691} a^{7} + \frac{12303195090928235359099892524102450112285807651}{33148933519983126522663205680116929888449877382} a^{6} + \frac{5990946287309577351582652661738420592539429750}{16574466759991563261331602840058464944224938691} a^{5} + \frac{8786051357789070948886281176460193991545969431}{33148933519983126522663205680116929888449877382} a^{4} + \frac{3502924746268302149028762424584079776428158864}{16574466759991563261331602840058464944224938691} a^{3} - \frac{316203388186891210359661480368212297825479713}{16574466759991563261331602840058464944224938691} a^{2} + \frac{3863160371072353106956077542081906289130672826}{16574466759991563261331602840058464944224938691} a + \frac{3075746592618522931036153146108241495917358773}{33148933519983126522663205680116929888449877382}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{12}$, which has order $144$ (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:  \( 207566.462553 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 8.8.1340095640625.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
7Data not computed