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Properties

Label 16.4.26968313608...625.20
Degree $16$
Signature $[4, 6]$
Discriminant $5^{12}\cdot 101^{10}$
Root discriminant $59.83$
Ramified primes $5, 101$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T875

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2861, -31859, 149429, -375564, 541710, -448959, 206202, -45982, -3507, 7578, -2578, 717, -66, 12, 13, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 13*x^14 + 12*x^13 - 66*x^12 + 717*x^11 - 2578*x^10 + 7578*x^9 - 3507*x^8 - 45982*x^7 + 206202*x^6 - 448959*x^5 + 541710*x^4 - 375564*x^3 + 149429*x^2 - 31859*x + 2861)
 
gp: K = bnfinit(x^16 - 3*x^15 + 13*x^14 + 12*x^13 - 66*x^12 + 717*x^11 - 2578*x^10 + 7578*x^9 - 3507*x^8 - 45982*x^7 + 206202*x^6 - 448959*x^5 + 541710*x^4 - 375564*x^3 + 149429*x^2 - 31859*x + 2861, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 13 x^{14} + 12 x^{13} - 66 x^{12} + 717 x^{11} - 2578 x^{10} + 7578 x^{9} - 3507 x^{8} - 45982 x^{7} + 206202 x^{6} - 448959 x^{5} + 541710 x^{4} - 375564 x^{3} + 149429 x^{2} - 31859 x + 2861 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26968313608671985107666015625=5^{12}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.83$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{500} a^{12} - \frac{21}{250} a^{11} - \frac{1}{10} a^{10} - \frac{7}{100} a^{9} + \frac{59}{125} a^{8} + \frac{7}{25} a^{7} - \frac{59}{500} a^{6} - \frac{4}{25} a^{5} - \frac{6}{125} a^{4} - \frac{41}{100} a^{3} + \frac{3}{10} a^{2} + \frac{3}{125} a - \frac{189}{500}$, $\frac{1}{500} a^{13} - \frac{7}{250} a^{11} - \frac{7}{100} a^{10} + \frac{83}{250} a^{9} + \frac{38}{125} a^{8} + \frac{21}{500} a^{7} + \frac{121}{250} a^{6} + \frac{29}{125} a^{5} - \frac{213}{500} a^{4} - \frac{8}{25} a^{3} + \frac{3}{125} a^{2} + \frac{23}{100} a - \frac{69}{250}$, $\frac{1}{32500} a^{14} - \frac{1}{6500} a^{13} - \frac{2}{8125} a^{12} - \frac{2417}{32500} a^{11} + \frac{141}{32500} a^{10} - \frac{2122}{8125} a^{9} - \frac{12723}{32500} a^{8} + \frac{2677}{32500} a^{7} + \frac{2463}{8125} a^{6} - \frac{8273}{32500} a^{5} - \frac{1739}{32500} a^{4} - \frac{1559}{16250} a^{3} + \frac{1031}{6500} a^{2} - \frac{11341}{32500} a - \frac{661}{8125}$, $\frac{1}{16534416755089669182500} a^{15} + \frac{69373192421395383}{16534416755089669182500} a^{14} + \frac{1575004346762868691}{8267208377544834591250} a^{13} + \frac{4000713402431586436}{4133604188772417295625} a^{12} + \frac{321607146375186105219}{3306883351017933836500} a^{11} + \frac{10728969244218994071}{826720837754483459125} a^{10} - \frac{3005552964011234452031}{8267208377544834591250} a^{9} + \frac{8676559184932563147}{33135103717614567500} a^{8} - \frac{3484584303337692401721}{8267208377544834591250} a^{7} - \frac{1170765453593259936868}{4133604188772417295625} a^{6} - \frac{4187882591827513518083}{16534416755089669182500} a^{5} + \frac{10804842830610265579}{33068833510179338365} a^{4} + \frac{816811578008435941623}{8267208377544834591250} a^{3} - \frac{1231379141175382584491}{16534416755089669182500} a^{2} + \frac{4035909341377740647589}{8267208377544834591250} a - \frac{2982841242748215432497}{16534416755089669182500}$

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:  $9$
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:  \( 109382898.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T875:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n875
Character table for t16n875 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.4.1625943765625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
101Data not computed