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Properties

Label 16.4.15231185783...001.14
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 67^{6}$
Root discriminant $57.73$
Ramified primes $17, 67$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^5.C_2.C_2$ (as 16T257)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10267, -16757, 19611, 21643, -25504, -7723, 6980, -10770, 57, 3942, -299, -354, -225, 110, -10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 10*x^14 + 110*x^13 - 225*x^12 - 354*x^11 - 299*x^10 + 3942*x^9 + 57*x^8 - 10770*x^7 + 6980*x^6 - 7723*x^5 - 25504*x^4 + 21643*x^3 + 19611*x^2 - 16757*x - 10267)
 
gp: K = bnfinit(x^16 - 3*x^15 - 10*x^14 + 110*x^13 - 225*x^12 - 354*x^11 - 299*x^10 + 3942*x^9 + 57*x^8 - 10770*x^7 + 6980*x^6 - 7723*x^5 - 25504*x^4 + 21643*x^3 + 19611*x^2 - 16757*x - 10267, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 10 x^{14} + 110 x^{13} - 225 x^{12} - 354 x^{11} - 299 x^{10} + 3942 x^{9} + 57 x^{8} - 10770 x^{7} + 6980 x^{6} - 7723 x^{5} - 25504 x^{4} + 21643 x^{3} + 19611 x^{2} - 16757 x - 10267 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15231185783695887615175635001=17^{14}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.73$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{208} a^{14} + \frac{7}{208} a^{13} - \frac{1}{16} a^{12} - \frac{11}{208} a^{11} - \frac{5}{104} a^{10} - \frac{15}{208} a^{9} + \frac{73}{208} a^{8} - \frac{5}{208} a^{7} - \frac{9}{104} a^{6} - \frac{29}{208} a^{5} - \frac{1}{52} a^{4} + \frac{37}{104} a^{3} - \frac{21}{52} a^{2} + \frac{9}{208} a - \frac{11}{208}$, $\frac{1}{281572301210249881982228960010688} a^{15} + \frac{11640650663809627534650344113}{8799134412820308811944655000334} a^{14} + \frac{9137931832726394850847870905859}{140786150605124940991114480005344} a^{13} - \frac{1312066529646389548191726290109}{17598268825640617623889310000668} a^{12} + \frac{13843816584070722593552577195471}{281572301210249881982228960010688} a^{11} - \frac{20446841485772113007090630294485}{281572301210249881982228960010688} a^{10} - \frac{6214274325243203105599130738741}{140786150605124940991114480005344} a^{9} + \frac{34019938345372459435788347213}{35196537651281235247778620001336} a^{8} + \frac{88564131289526117497665039268481}{281572301210249881982228960010688} a^{7} - \frac{132010686381902499649916445606127}{281572301210249881982228960010688} a^{6} - \frac{41950531731361522727802655709273}{281572301210249881982228960010688} a^{5} - \frac{29940825032376961902539674011451}{140786150605124940991114480005344} a^{4} - \frac{53298473470281290536950095930169}{140786150605124940991114480005344} a^{3} - \frac{13123032635221488516466290570475}{281572301210249881982228960010688} a^{2} + \frac{5425368304272953578736154295981}{140786150605124940991114480005344} a + \frac{94150824345060682263192117331945}{281572301210249881982228960010688}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $9$
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:  \( 7218829.00139 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^5.C_2.C_2$ (as 16T257):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5.C_2.C_2$
Character table for $C_2^5.C_2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.1842010303097.2, 8.2.27492691091.1, 8.6.7259687665147.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
67Data not computed