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Properties

Label 16.0.10758673145...4096.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 223^{8}$
Root discriminant $100.46$
Ramified primes $2, 223$
Class number $138016$ (GRH)
Class group $[2, 2, 2, 2, 8626]$ (GRH)
Galois group $C_2^3:S_4.C_2$ (as 16T764)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![530507567, 125219124, 464127720, -21775820, 325754186, -121487192, 104949032, -29764972, 11890001, -2252232, 610208, -77252, 15852, -1260, 204, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 204*x^14 - 1260*x^13 + 15852*x^12 - 77252*x^11 + 610208*x^10 - 2252232*x^9 + 11890001*x^8 - 29764972*x^7 + 104949032*x^6 - 121487192*x^5 + 325754186*x^4 - 21775820*x^3 + 464127720*x^2 + 125219124*x + 530507567)
 
gp: K = bnfinit(x^16 - 8*x^15 + 204*x^14 - 1260*x^13 + 15852*x^12 - 77252*x^11 + 610208*x^10 - 2252232*x^9 + 11890001*x^8 - 29764972*x^7 + 104949032*x^6 - 121487192*x^5 + 325754186*x^4 - 21775820*x^3 + 464127720*x^2 + 125219124*x + 530507567, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 204 x^{14} - 1260 x^{13} + 15852 x^{12} - 77252 x^{11} + 610208 x^{10} - 2252232 x^{9} + 11890001 x^{8} - 29764972 x^{7} + 104949032 x^{6} - 121487192 x^{5} + 325754186 x^{4} - 21775820 x^{3} + 464127720 x^{2} + 125219124 x + 530507567 \)

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:  \(107586731453760569991937021444096=2^{44}\cdot 223^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $100.46$
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:  $2, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{8}{19} a^{12} + \frac{3}{19} a^{11} - \frac{3}{19} a^{10} + \frac{7}{19} a^{9} + \frac{8}{19} a^{8} - \frac{8}{19} a^{7} - \frac{9}{19} a^{6} + \frac{7}{19} a^{5} - \frac{9}{19} a^{4} + \frac{9}{19} a^{3} + \frac{3}{19} a^{2} + \frac{4}{19} a + \frac{4}{19}$, $\frac{1}{95} a^{14} + \frac{2}{95} a^{13} - \frac{39}{95} a^{12} - \frac{11}{95} a^{11} + \frac{34}{95} a^{10} + \frac{21}{95} a^{9} - \frac{4}{95} a^{8} + \frac{5}{19} a^{7} - \frac{7}{95} a^{6} + \frac{23}{95} a^{5} + \frac{33}{95} a^{4} - \frac{21}{95} a^{3} - \frac{23}{95} a^{2} - \frac{32}{95} a - \frac{36}{95}$, $\frac{1}{753332536235007355586557321532622762555744114120122094738485004845} a^{15} - \frac{537294925529149827076280679428625483022410834162604729489306664}{150666507247001471117311464306524552511148822824024418947697000969} a^{14} + \frac{638813653068346882868099025287165923100882645168051910696173853}{39649080854474071346660911659611724345039163901059057617815000255} a^{13} - \frac{319618688089750185052880977558087378049349783203322023476930979963}{753332536235007355586557321532622762555744114120122094738485004845} a^{12} - \frac{147949877918763758185440698244188747622404696262004929073657653779}{753332536235007355586557321532622762555744114120122094738485004845} a^{11} - \frac{3729692516037113701975644907382274146840648525807672779523299517}{753332536235007355586557321532622762555744114120122094738485004845} a^{10} + \frac{180864451821241129850584720249852178798622344984767795336851845764}{753332536235007355586557321532622762555744114120122094738485004845} a^{9} + \frac{299569326370950689374353975528284483754058116283312282589585472938}{753332536235007355586557321532622762555744114120122094738485004845} a^{8} + \frac{277804753028239523717757457431282429605253645418569319513967038933}{753332536235007355586557321532622762555744114120122094738485004845} a^{7} + \frac{30722429976190597329218218379630840858750656676136665735998633362}{753332536235007355586557321532622762555744114120122094738485004845} a^{6} - \frac{370004563647403406468069845508645001660288770233888262534426000983}{753332536235007355586557321532622762555744114120122094738485004845} a^{5} - \frac{102161304532318968231496742037295396005029631076171152893637835642}{753332536235007355586557321532622762555744114120122094738485004845} a^{4} - \frac{361107340848816278959820007931382979313142200050995252428478484236}{753332536235007355586557321532622762555744114120122094738485004845} a^{3} - \frac{191435731909530454918346335551803471318728398680545558451362984591}{753332536235007355586557321532622762555744114120122094738485004845} a^{2} - \frac{196048278208836602362655995072883380640560155768925185621969931192}{753332536235007355586557321532622762555744114120122094738485004845} a + \frac{124480796766629676664670831848433125918283837577207637607352761927}{753332536235007355586557321532622762555744114120122094738485004845}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{8626}$, which has order $138016$ (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:  \( 14766.512467 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3:S_4.C_2$ (as 16T764):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 384
The 23 conjugacy class representatives for $C_2^3:S_4.C_2$
Character table for $C_2^3:S_4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14272.1, 8.8.3259039744.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
2Data not computed
223Data not computed