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Properties

Label 16.16.1075867314...4096.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 223^{8}$
Root discriminant $100.46$
Ramified primes $2, 223$
Class number $1$ (GRH)
Class group Trivial (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![16653503, -92368028, -284134756, -127064324, 161682534, 105425840, -33317060, -23144124, 4122561, 1952144, -287260, -73564, 10036, 1260, -164, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 164*x^14 + 1260*x^13 + 10036*x^12 - 73564*x^11 - 287260*x^10 + 1952144*x^9 + 4122561*x^8 - 23144124*x^7 - 33317060*x^6 + 105425840*x^5 + 161682534*x^4 - 127064324*x^3 - 284134756*x^2 - 92368028*x + 16653503)
 
gp: K = bnfinit(x^16 - 8*x^15 - 164*x^14 + 1260*x^13 + 10036*x^12 - 73564*x^11 - 287260*x^10 + 1952144*x^9 + 4122561*x^8 - 23144124*x^7 - 33317060*x^6 + 105425840*x^5 + 161682534*x^4 - 127064324*x^3 - 284134756*x^2 - 92368028*x + 16653503, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 164 x^{14} + 1260 x^{13} + 10036 x^{12} - 73564 x^{11} - 287260 x^{10} + 1952144 x^{9} + 4122561 x^{8} - 23144124 x^{7} - 33317060 x^{6} + 105425840 x^{5} + 161682534 x^{4} - 127064324 x^{3} - 284134756 x^{2} - 92368028 x + 16653503 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5410971914729521707864423564661233322869403554259924121909} a^{15} - \frac{505077532329456627439015796147617009705268820405353067208}{5410971914729521707864423564661233322869403554259924121909} a^{14} - \frac{1573674822927437235133709318591953960663729626544080401306}{5410971914729521707864423564661233322869403554259924121909} a^{13} - \frac{254692250917555038201996389693375251128491199488128909962}{5410971914729521707864423564661233322869403554259924121909} a^{12} - \frac{1217600457837261069072533957274835365188952963997731626889}{5410971914729521707864423564661233322869403554259924121909} a^{11} + \frac{737478594380339056751866844130649815199214822756055902448}{5410971914729521707864423564661233322869403554259924121909} a^{10} - \frac{409567351556136229154613088800229105601039210422650829239}{5410971914729521707864423564661233322869403554259924121909} a^{9} - \frac{1851894707422615352034570979155681472715435131502491038241}{5410971914729521707864423564661233322869403554259924121909} a^{8} + \frac{864430363230094777819810525229121293529253647786584699007}{5410971914729521707864423564661233322869403554259924121909} a^{7} - \frac{2475806864785901592131882911109142400693917976873975883138}{5410971914729521707864423564661233322869403554259924121909} a^{6} + \frac{1444927335900900144362758304631780749896693706958650827959}{5410971914729521707864423564661233322869403554259924121909} a^{5} + \frac{713137826976429743881228531465268040357689185328644278516}{5410971914729521707864423564661233322869403554259924121909} a^{4} + \frac{506918346580344572591379026092349276495346217001607655952}{5410971914729521707864423564661233322869403554259924121909} a^{3} + \frac{2073696607036406739427171894559303108270942837521524067229}{5410971914729521707864423564661233322869403554259924121909} a^{2} + \frac{1431710697595262912695546331039187504427302732001129160264}{5410971914729521707864423564661233322869403554259924121909} a - \frac{2193054808611149378209665484647572019895016355505340482288}{5410971914729521707864423564661233322869403554259924121909}$

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:  $15$
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:  \( 66964254976.1 \) (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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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