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Properties

Label 16.8.25557302165...0625.4
Degree $16$
Signature $[8, 4]$
Discriminant $5^{10}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}$
Root discriminant $163.29$
Ramified primes $5, 29, 89, 97$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![217181440849, 49037720024, -116425745573, 7085774092, 13345340432, -3537270704, 458923898, 59677972, -34332018, 4385924, -936609, -16080, 5384, -508, 212, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 212*x^14 - 508*x^13 + 5384*x^12 - 16080*x^11 - 936609*x^10 + 4385924*x^9 - 34332018*x^8 + 59677972*x^7 + 458923898*x^6 - 3537270704*x^5 + 13345340432*x^4 + 7085774092*x^3 - 116425745573*x^2 + 49037720024*x + 217181440849)
 
gp: K = bnfinit(x^16 + 212*x^14 - 508*x^13 + 5384*x^12 - 16080*x^11 - 936609*x^10 + 4385924*x^9 - 34332018*x^8 + 59677972*x^7 + 458923898*x^6 - 3537270704*x^5 + 13345340432*x^4 + 7085774092*x^3 - 116425745573*x^2 + 49037720024*x + 217181440849, 1)
 

Normalized defining polynomial

\( x^{16} + 212 x^{14} - 508 x^{13} + 5384 x^{12} - 16080 x^{11} - 936609 x^{10} + 4385924 x^{9} - 34332018 x^{8} + 59677972 x^{7} + 458923898 x^{6} - 3537270704 x^{5} + 13345340432 x^{4} + 7085774092 x^{3} - 116425745573 x^{2} + 49037720024 x + 217181440849 \)

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:  \(255573021657714643907277003525390625=5^{10}\cdot 29^{6}\cdot 89^{6}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $163.29$
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:  $5, 29, 89, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{3}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{3}{16} a^{5} - \frac{5}{16} a^{4} + \frac{3}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{16} a + \frac{1}{16}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{10} + \frac{1}{16} a^{9} + \frac{11}{64} a^{8} - \frac{1}{8} a^{7} - \frac{1}{64} a^{6} - \frac{1}{4} a^{5} + \frac{1}{64} a^{4} - \frac{1}{16} a^{3} - \frac{19}{64} a^{2} - \frac{1}{8} a - \frac{23}{64}$, $\frac{1}{256} a^{13} + \frac{1}{256} a^{12} - \frac{1}{128} a^{11} + \frac{1}{128} a^{10} - \frac{49}{256} a^{9} + \frac{35}{256} a^{8} - \frac{41}{256} a^{7} - \frac{49}{256} a^{6} - \frac{15}{256} a^{5} - \frac{67}{256} a^{4} + \frac{105}{256} a^{3} - \frac{27}{256} a^{2} + \frac{65}{256} a - \frac{87}{256}$, $\frac{1}{1024} a^{14} - \frac{1}{512} a^{13} - \frac{5}{1024} a^{12} + \frac{1}{128} a^{11} - \frac{55}{1024} a^{10} + \frac{27}{512} a^{9} + \frac{55}{512} a^{8} - \frac{27}{512} a^{7} - \frac{31}{256} a^{6} + \frac{53}{512} a^{5} + \frac{153}{512} a^{4} - \frac{171}{512} a^{3} - \frac{247}{512} a^{2} - \frac{77}{512} a - \frac{123}{1024}$, $\frac{1}{37351621922516400041045920677597190398449156425611113693697092442929313918976} a^{15} - \frac{17114371743962543471297414236535173780079851012058848591918020137551598379}{37351621922516400041045920677597190398449156425611113693697092442929313918976} a^{14} - \frac{71847951998922185045676745742115824578750626488744596487923570214610670755}{37351621922516400041045920677597190398449156425611113693697092442929313918976} a^{13} + \frac{124742649724316441611862426902203781237191208121575527624164403936087039109}{37351621922516400041045920677597190398449156425611113693697092442929313918976} a^{12} + \frac{923086166452788624544999326157578646766648359774617051131106853169093228385}{37351621922516400041045920677597190398449156425611113693697092442929313918976} a^{11} + \frac{221972375176265972792483898178317149794936272138257180170416434279595026917}{37351621922516400041045920677597190398449156425611113693697092442929313918976} a^{10} + \frac{391079274440477317220857259933127260318212121447624764758174157638302671719}{4668952740314550005130740084699648799806144553201389211712136555366164239872} a^{9} + \frac{1046375603079369018480475591458978528694272019983378731502316539518268843151}{9337905480629100010261480169399297599612289106402778423424273110732328479744} a^{8} + \frac{3135996091119903558385307813121800251335722114569136971273623168228581784781}{18675810961258200020522960338798595199224578212805556846848546221464656959488} a^{7} + \frac{3840167347852933863650830187916287115593614663204934720602460132299752178763}{18675810961258200020522960338798595199224578212805556846848546221464656959488} a^{6} - \frac{219153907633798449137068437446890606519199448772590839175954479182042429079}{4668952740314550005130740084699648799806144553201389211712136555366164239872} a^{5} - \frac{2163651175165332709692042282572032321184779935551709978187509657206978446749}{4668952740314550005130740084699648799806144553201389211712136555366164239872} a^{4} + \frac{1265520915116709678929612231010997780560680494275650231486715224468088612349}{4668952740314550005130740084699648799806144553201389211712136555366164239872} a^{3} - \frac{2434148126415518597656434781150920606563159255246405993868099465352328979395}{9337905480629100010261480169399297599612289106402778423424273110732328479744} a^{2} - \frac{6986663156208847711138012969127707575460149875608488423032909555559914336961}{37351621922516400041045920677597190398449156425611113693697092442929313918976} a - \frac{13171973904755828225386262823408269140961406435644142785082030032741133904413}{37351621922516400041045920677597190398449156425611113693697092442929313918976}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 320078741386 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2225.1, 4.4.725.1, 4.4.64525.1, 8.8.4163475625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.4$x^{4} + 232$$4$$1$$3$$C_4$$[\ ]_{4}$
$89$89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.2.1.2$x^{2} + 267$$2$$1$$1$$C_2$$[\ ]_{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$