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Properties

Label 16.0.64000000000...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{34}\cdot 5^{28}$
Root discriminant $72.93$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $A_{16}$ (as 16T1953)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169955, -563280, 822650, -700350, 366350, -95116, -1494, 7670, 2940, -5080, 3262, -1522, 530, -140, 30, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 30*x^14 - 140*x^13 + 530*x^12 - 1522*x^11 + 3262*x^10 - 5080*x^9 + 2940*x^8 + 7670*x^7 - 1494*x^6 - 95116*x^5 + 366350*x^4 - 700350*x^3 + 822650*x^2 - 563280*x + 169955)
 
gp: K = bnfinit(x^16 - 6*x^15 + 30*x^14 - 140*x^13 + 530*x^12 - 1522*x^11 + 3262*x^10 - 5080*x^9 + 2940*x^8 + 7670*x^7 - 1494*x^6 - 95116*x^5 + 366350*x^4 - 700350*x^3 + 822650*x^2 - 563280*x + 169955, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 30 x^{14} - 140 x^{13} + 530 x^{12} - 1522 x^{11} + 3262 x^{10} - 5080 x^{9} + 2940 x^{8} + 7670 x^{7} - 1494 x^{6} - 95116 x^{5} + 366350 x^{4} - 700350 x^{3} + 822650 x^{2} - 563280 x + 169955 \)

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:  \(640000000000000000000000000000=2^{34}\cdot 5^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.93$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{16} a^{2} + \frac{3}{8} a + \frac{3}{16}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} + \frac{11}{32} a^{3} - \frac{3}{32} a^{2} - \frac{9}{32} a + \frac{11}{32}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{9} + \frac{1}{64} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} + \frac{15}{64} a^{4} - \frac{1}{8} a^{3} + \frac{7}{16} a^{2} + \frac{3}{32} a + \frac{15}{64}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{1}{64} a^{10} + \frac{3}{128} a^{9} + \frac{7}{128} a^{8} + \frac{3}{32} a^{7} + \frac{1}{16} a^{6} + \frac{3}{128} a^{5} - \frac{23}{128} a^{4} - \frac{7}{32} a^{3} - \frac{11}{64} a^{2} - \frac{55}{128} a - \frac{31}{128}$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{12} - \frac{1}{64} a^{11} + \frac{1}{128} a^{10} - \frac{3}{64} a^{9} + \frac{3}{128} a^{8} - \frac{3}{32} a^{7} + \frac{11}{128} a^{6} + \frac{3}{32} a^{5} + \frac{13}{128} a^{4} - \frac{9}{64} a^{3} + \frac{51}{128} a^{2} + \frac{13}{64} a + \frac{49}{128}$, $\frac{1}{20638000254448367104206080} a^{15} - \frac{916801925271557749631}{4127600050889673420841216} a^{14} - \frac{7024026324957938248525}{4127600050889673420841216} a^{13} + \frac{19352792449932544254461}{4127600050889673420841216} a^{12} - \frac{49864308430994891173149}{4127600050889673420841216} a^{11} - \frac{210454449843252304598857}{20638000254448367104206080} a^{10} - \frac{49676439315584366527223}{4127600050889673420841216} a^{9} - \frac{245289271519301309178489}{4127600050889673420841216} a^{8} - \frac{140972087516731269832765}{4127600050889673420841216} a^{7} + \frac{357089662670275350606855}{4127600050889673420841216} a^{6} - \frac{4985388541522238338598519}{20638000254448367104206080} a^{5} - \frac{188924054321568729399093}{4127600050889673420841216} a^{4} + \frac{359370425820146703625069}{4127600050889673420841216} a^{3} + \frac{1387257258331620509104013}{4127600050889673420841216} a^{2} + \frac{1432334250288065296814255}{4127600050889673420841216} a + \frac{2061849615975710175360017}{4127600050889673420841216}$

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:  $7$
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:  \( 2085624086.88 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_{16}$ (as 16T1953):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 10461394944000
The 123 conjugacy class representatives for $A_{16}$ are not computed
Character table for $A_{16}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.13.0.1}{13} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $15{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.4.8.8$x^{4} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.8.22.28$x^{8} + 2 x^{4} + 8 x^{2} + 4$$4$$2$$22$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.9.2$x^{5} + 55$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.1$x^{10} + 5$$10$$1$$19$$F_5$$[9/4]_{4}$