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Properties

Label 16.0.69019132572...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{6}\cdot 101^{6}\cdot 409^{4}$
Root discriminant $200.65$
Ramified primes $5, 29, 101, 409$
Class number $768$ (GRH)
Class group $[2, 2, 4, 48]$ (GRH)
Galois group 16T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5083593679, 13763104, -563128918, -369460016, 300846774, -57678898, -6719053, 1065390, 888942, -143182, 2907, -3862, 666, -256, 97, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 97*x^14 - 256*x^13 + 666*x^12 - 3862*x^11 + 2907*x^10 - 143182*x^9 + 888942*x^8 + 1065390*x^7 - 6719053*x^6 - 57678898*x^5 + 300846774*x^4 - 369460016*x^3 - 563128918*x^2 + 13763104*x + 5083593679)
 
gp: K = bnfinit(x^16 - 4*x^15 + 97*x^14 - 256*x^13 + 666*x^12 - 3862*x^11 + 2907*x^10 - 143182*x^9 + 888942*x^8 + 1065390*x^7 - 6719053*x^6 - 57678898*x^5 + 300846774*x^4 - 369460016*x^3 - 563128918*x^2 + 13763104*x + 5083593679, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 97 x^{14} - 256 x^{13} + 666 x^{12} - 3862 x^{11} + 2907 x^{10} - 143182 x^{9} + 888942 x^{8} + 1065390 x^{7} - 6719053 x^{6} - 57678898 x^{5} + 300846774 x^{4} - 369460016 x^{3} - 563128918 x^{2} + 13763104 x + 5083593679 \)

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:  \(6901913257237170531380177586750390625=5^{8}\cdot 29^{6}\cdot 101^{6}\cdot 409^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.65$
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:  $5, 29, 101, 409$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{3775679251329447713912971069863439822017870413109911508266226041059082} a^{15} + \frac{52462792939908228621783870800941642360020672644519721387356499412687}{1887839625664723856956485534931719911008935206554955754133113020529541} a^{14} - \frac{453439135651755374200089499391896788171232523974055832413597091022124}{1887839625664723856956485534931719911008935206554955754133113020529541} a^{13} + \frac{228924910773207126239124280951814521459855335741323906391346568148951}{3775679251329447713912971069863439822017870413109911508266226041059082} a^{12} - \frac{670423578825742232123152354320381874801593382640260158301180322060455}{3775679251329447713912971069863439822017870413109911508266226041059082} a^{11} - \frac{157279876342100325297240761681739520486297825370560038564837036472222}{1887839625664723856956485534931719911008935206554955754133113020529541} a^{10} + \frac{353343543293400334343453746310912830298772066130567664533397624659949}{3775679251329447713912971069863439822017870413109911508266226041059082} a^{9} + \frac{18006159730897667098208348188709121844320958777509215616194852510879}{343243568302677064901179188169403620183442764828173773478747821914462} a^{8} + \frac{317325057267441567141783095533559137644889411944650475109322342314596}{1887839625664723856956485534931719911008935206554955754133113020529541} a^{7} - \frac{264522040380313964451496213825411553833683246861447358150773106715154}{1887839625664723856956485534931719911008935206554955754133113020529541} a^{6} - \frac{1546608257224338317056711721853950671985742635459427862815666172849629}{3775679251329447713912971069863439822017870413109911508266226041059082} a^{5} + \frac{1817591222719208310544285671572959857816725067333076485276899495550115}{3775679251329447713912971069863439822017870413109911508266226041059082} a^{4} - \frac{706848970517659093933462427114058283619950872933709256637618129952241}{1887839625664723856956485534931719911008935206554955754133113020529541} a^{3} - \frac{1825158574128142855494945673677459222088012002124634563028183891522365}{3775679251329447713912971069863439822017870413109911508266226041059082} a^{2} + \frac{800856867808711776048672411466447809254535916934974569715828587806341}{1887839625664723856956485534931719911008935206554955754133113020529541} a - \frac{90334968668647757673586484766167904610757238227188962273784323761965}{3775679251329447713912971069863439822017870413109911508266226041059082}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{48}$, which has order $768$ (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:  \( 419559822.645 \) (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

\(\Q(\sqrt{5}) \), 4.4.73225.2, 4.0.1032725.1, 4.0.296525.2, 8.0.896944098450625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.8.4.1$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$29.2.0.1$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}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
409Data not computed