↑

Properties

Label 16.8.42905375783...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 29^{6}\cdot 41^{4}\cdot 79^{4}$
Root discriminant $168.67$
Ramified primes $2, 5, 29, 41, 79$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T839

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![263810509, 401578682, -414853388, 74479794, -10427857, 8160446, 3104020, -2942444, 800340, -162364, 14124, 4828, -1098, 270, -36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 36*x^14 + 270*x^13 - 1098*x^12 + 4828*x^11 + 14124*x^10 - 162364*x^9 + 800340*x^8 - 2942444*x^7 + 3104020*x^6 + 8160446*x^5 - 10427857*x^4 + 74479794*x^3 - 414853388*x^2 + 401578682*x + 263810509)
 
gp: K = bnfinit(x^16 - 6*x^15 - 36*x^14 + 270*x^13 - 1098*x^12 + 4828*x^11 + 14124*x^10 - 162364*x^9 + 800340*x^8 - 2942444*x^7 + 3104020*x^6 + 8160446*x^5 - 10427857*x^4 + 74479794*x^3 - 414853388*x^2 + 401578682*x + 263810509, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 36 x^{14} + 270 x^{13} - 1098 x^{12} + 4828 x^{11} + 14124 x^{10} - 162364 x^{9} + 800340 x^{8} - 2942444 x^{7} + 3104020 x^{6} + 8160446 x^{5} - 10427857 x^{4} + 74479794 x^{3} - 414853388 x^{2} + 401578682 x + 263810509 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(429053757833296459233761689600000000=2^{24}\cdot 5^{8}\cdot 29^{6}\cdot 41^{4}\cdot 79^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.67$
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:  $2, 5, 29, 41, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{31} a^{14} + \frac{5}{31} a^{13} - \frac{3}{31} a^{12} + \frac{3}{31} a^{11} - \frac{7}{31} a^{10} + \frac{4}{31} a^{9} - \frac{12}{31} a^{7} + \frac{5}{31} a^{6} - \frac{8}{31} a^{5} + \frac{9}{31} a^{4} + \frac{2}{31} a^{3} - \frac{5}{31} a^{2} - \frac{6}{31} a + \frac{2}{31}$, $\frac{1}{18846271003250824554641462197494325545292976496937241106831083} a^{15} + \frac{53906867978020609146470474992769962086850515726100455360}{1713297363931893141331042017954029595026634226994294646075553} a^{14} - \frac{3576697224315110170292697858679191322627713348991040891994622}{18846271003250824554641462197494325545292976496937241106831083} a^{13} - \frac{78319669235826213485120847827462042870560660776625114429524}{18846271003250824554641462197494325545292976496937241106831083} a^{12} + \frac{4161838341054178500500176146814575977936053413679270053647576}{18846271003250824554641462197494325545292976496937241106831083} a^{11} + \frac{3935519336803491379672464568669596136492442374413078524585849}{18846271003250824554641462197494325545292976496937241106831083} a^{10} - \frac{5378953019703300492198976204032503207876900885118474621590186}{18846271003250824554641462197494325545292976496937241106831083} a^{9} + \frac{851132166615266087878448670721595492199760725390845597840072}{2692324429035832079234494599642046506470425213848177300975869} a^{8} + \frac{587494381641870205323596807529533780770544578072387000355405}{1449713154096217273433958630576486580407152038225941623602391} a^{7} - \frac{1602353145048360839921344506350689402370026698023739732309399}{18846271003250824554641462197494325545292976496937241106831083} a^{6} - \frac{4217076993468298231113876652912481364859987713012565738592172}{18846271003250824554641462197494325545292976496937241106831083} a^{5} + \frac{1273678673272000252884588172974446739157512913099262211169178}{18846271003250824554641462197494325545292976496937241106831083} a^{4} + \frac{572128082746722151122308557209530277844805993572508337061454}{18846271003250824554641462197494325545292976496937241106831083} a^{3} - \frac{3690849939311128136580277877791942337811654024420868778830464}{18846271003250824554641462197494325545292976496937241106831083} a^{2} + \frac{445857262652471624743624119802430717903298566247135041676888}{1713297363931893141331042017954029595026634226994294646075553} a - \frac{4398243717063290757092709193507531537091757508248013439850894}{18846271003250824554641462197494325545292976496937241106831083}$

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:  $11$
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:  \( 904508068119 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n839 are not computed
Character table for t16n839 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.916400.1, 4.4.31600.1, 4.4.725.1, 8.8.839788960000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$79$$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{79}$$x + 2$$1$$1$$0$Trivial$[\ ]$
79.8.4.1$x^{8} + 37446 x^{4} - 493039 x^{2} + 350550729$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$