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Properties

Label 16.8.80073733711...8449.1
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 73^{14}$
Root discriminant $73.96$
Ramified primes $3, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25408, -2001824, -5355776, -7290744, -6547056, -3438310, -570157, 353057, 181451, -212, -15701, -1001, 1033, 80, -47, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408)
 
gp: K = bnfinit(x^16 - 3*x^15 - 47*x^14 + 80*x^13 + 1033*x^12 - 1001*x^11 - 15701*x^10 - 212*x^9 + 181451*x^8 + 353057*x^7 - 570157*x^6 - 3438310*x^5 - 6547056*x^4 - 7290744*x^3 - 5355776*x^2 - 2001824*x + 25408, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 47 x^{14} + 80 x^{13} + 1033 x^{12} - 1001 x^{11} - 15701 x^{10} - 212 x^{9} + 181451 x^{8} + 353057 x^{7} - 570157 x^{6} - 3438310 x^{5} - 6547056 x^{4} - 7290744 x^{3} - 5355776 x^{2} - 2001824 x + 25408 \)

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:  \(800737337114777368288115578449=3^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.96$
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:  $3, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{5}{16} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{9} - \frac{1}{32} a^{7} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{32} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{8} + \frac{1}{8} a^{5} - \frac{7}{32} a^{4} + \frac{1}{8} a^{3} + \frac{5}{16} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{9} - \frac{1}{8} a^{6} - \frac{7}{32} a^{5} - \frac{1}{8} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{512} a^{14} - \frac{1}{512} a^{13} - \frac{5}{512} a^{12} - \frac{3}{256} a^{11} - \frac{15}{512} a^{10} - \frac{15}{512} a^{9} + \frac{25}{512} a^{8} - \frac{11}{256} a^{7} + \frac{27}{512} a^{6} - \frac{113}{512} a^{5} - \frac{75}{512} a^{4} - \frac{1}{64} a^{3} - \frac{37}{128} a^{2} - \frac{1}{8} a - \frac{3}{32}$, $\frac{1}{44635089381439681583271184918086656} a^{15} - \frac{30342772117526971823030433036803}{44635089381439681583271184918086656} a^{14} - \frac{609774044836455951252937420098003}{44635089381439681583271184918086656} a^{13} - \frac{133906841042647932083819532126371}{11158772345359920395817796229521664} a^{12} + \frac{163888426340017357663704612327741}{44635089381439681583271184918086656} a^{11} - \frac{434701297276275690519437398212241}{44635089381439681583271184918086656} a^{10} - \frac{1726549455936593700528402805812281}{44635089381439681583271184918086656} a^{9} + \frac{127933237917977871673471870431849}{5579386172679960197908898114760832} a^{8} - \frac{5010629326284151488938425227056697}{44635089381439681583271184918086656} a^{7} + \frac{5110186705342774899025650750612665}{44635089381439681583271184918086656} a^{6} + \frac{2015206467629447592071292383992839}{44635089381439681583271184918086656} a^{5} + \frac{3772424015131089878596633911298687}{22317544690719840791635592459043328} a^{4} - \frac{1707165559075027177103349504290905}{11158772345359920395817796229521664} a^{3} - \frac{29257082856839705872234636047483}{5579386172679960197908898114760832} a^{2} - \frac{204972837096164565768206702956475}{2789693086339980098954449057380416} a - \frac{304302550689292957880851632407773}{1394846543169990049477224528690208}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 9033509529.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{73}) \), 4.4.389017.1, 4.2.15987.1, 4.2.1167051.1, 8.4.99426586671873.1, 8.8.894839280046857.1, 8.4.1362008036601.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
73Data not computed