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Properties

Label 16.0.28596554320...5089.3
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 19^{8}$
Root discriminant $52.00$
Ramified primes $17, 19$
Class number $100$ (GRH)
Class group $[10, 10]$ (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![242827, -78291, 388080, -246727, 256578, -151578, 94489, -41394, 18122, -5341, 1336, -110, -39, 30, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827)
 
gp: K = bnfinit(x^16 - 2*x^15 - 8*x^14 + 30*x^13 - 39*x^12 - 110*x^11 + 1336*x^10 - 5341*x^9 + 18122*x^8 - 41394*x^7 + 94489*x^6 - 151578*x^5 + 256578*x^4 - 246727*x^3 + 388080*x^2 - 78291*x + 242827, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 8 x^{14} + 30 x^{13} - 39 x^{12} - 110 x^{11} + 1336 x^{10} - 5341 x^{9} + 18122 x^{8} - 41394 x^{7} + 94489 x^{6} - 151578 x^{5} + 256578 x^{4} - 246727 x^{3} + 388080 x^{2} - 78291 x + 242827 \)

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:  \(2859655432078149808865465089=17^{14}\cdot 19^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.00$
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:  $17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{202} a^{14} - \frac{1}{101} a^{13} - \frac{22}{101} a^{12} + \frac{1}{202} a^{11} + \frac{15}{101} a^{10} + \frac{28}{101} a^{9} - \frac{47}{202} a^{8} + \frac{8}{101} a^{7} + \frac{9}{101} a^{6} - \frac{55}{202} a^{5} + \frac{6}{101} a^{4} + \frac{42}{101} a^{3} - \frac{91}{202} a^{2} + \frac{11}{101} a + \frac{41}{101}$, $\frac{1}{30680715564551671347020486765663253998} a^{15} + \frac{32740667608595336515457479367274402}{15340357782275835673510243382831626999} a^{14} + \frac{735895873291543295192461032198829981}{15340357782275835673510243382831626999} a^{13} + \frac{2444859261728794114672214225329881811}{30680715564551671347020486765663253998} a^{12} + \frac{2308257514020303504021225449278769277}{15340357782275835673510243382831626999} a^{11} + \frac{3202765413852903218903998285938703106}{15340357782275835673510243382831626999} a^{10} - \frac{1904197533420227376396715555241496495}{30680715564551671347020486765663253998} a^{9} - \frac{3146413245660217320986449670808235299}{15340357782275835673510243382831626999} a^{8} + \frac{6139949456353416847588095946009696165}{15340357782275835673510243382831626999} a^{7} - \frac{13579780239124946376274544572556617265}{30680715564551671347020486765663253998} a^{6} - \frac{4470354528646966191622656503430211516}{15340357782275835673510243382831626999} a^{5} + \frac{3069571307753722995097560967851602706}{15340357782275835673510243382831626999} a^{4} + \frac{2826467812989772694473756715391511483}{30680715564551671347020486765663253998} a^{3} + \frac{7265287201608760442441158970512761007}{15340357782275835673510243382831626999} a^{2} - \frac{2289241656588876674278227091730029274}{15340357782275835673510243382831626999} a + \frac{6899002628511704700509004066951738436}{15340357782275835673510243382831626999}$

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:  \( 149969.889475 \) (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

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.5491.1, 4.2.93347.1, 8.0.53475746204033.2, 8.4.148132260953.1, 8.4.8713662409.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
17Data not computed
$19$19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.8.4.1$x^{8} + 7220 x^{4} - 27436 x^{2} + 13032100$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$