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Properties

Label 16.2.32262667623...0000.1
Degree $16$
Signature $[2, 7]$
Discriminant $-\,2^{20}\cdot 5^{8}\cdot 31^{12}$
Root discriminant $69.87$
Ramified primes $2, 5, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18985591, -129895624, -55589321, -16826964, -5676142, -3753786, -927656, 238200, -30631, -17436, 15614, 580, -538, 126, -17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 17*x^14 + 126*x^13 - 538*x^12 + 580*x^11 + 15614*x^10 - 17436*x^9 - 30631*x^8 + 238200*x^7 - 927656*x^6 - 3753786*x^5 - 5676142*x^4 - 16826964*x^3 - 55589321*x^2 - 129895624*x - 18985591)
 
gp: K = bnfinit(x^16 - 6*x^15 - 17*x^14 + 126*x^13 - 538*x^12 + 580*x^11 + 15614*x^10 - 17436*x^9 - 30631*x^8 + 238200*x^7 - 927656*x^6 - 3753786*x^5 - 5676142*x^4 - 16826964*x^3 - 55589321*x^2 - 129895624*x - 18985591, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 17 x^{14} + 126 x^{13} - 538 x^{12} + 580 x^{11} + 15614 x^{10} - 17436 x^{9} - 30631 x^{8} + 238200 x^{7} - 927656 x^{6} - 3753786 x^{5} - 5676142 x^{4} - 16826964 x^{3} - 55589321 x^{2} - 129895624 x - 18985591 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-322626676239789982105600000000=-\,2^{20}\cdot 5^{8}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.87$
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:  $2, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} + \frac{1}{11} a^{12} + \frac{4}{11} a^{11} - \frac{2}{11} a^{10} + \frac{5}{11} a^{9} - \frac{5}{11} a^{8} - \frac{5}{11} a^{7} + \frac{1}{11} a^{6} - \frac{1}{11} a^{5} + \frac{5}{11} a^{4} + \frac{1}{11} a^{3} + \frac{2}{11} a^{2} + \frac{4}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{14} + \frac{3}{11} a^{12} + \frac{5}{11} a^{11} - \frac{4}{11} a^{10} + \frac{1}{11} a^{9} - \frac{5}{11} a^{7} - \frac{2}{11} a^{6} - \frac{5}{11} a^{5} - \frac{4}{11} a^{4} + \frac{1}{11} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a + \frac{4}{11}$, $\frac{1}{10641625903357416830693572717234701141841485273563233367471} a^{15} + \frac{157689741177540243228891515859974275348332722372970357109}{10641625903357416830693572717234701141841485273563233367471} a^{14} + \frac{187479385765351007317309142338637812906468231916999917908}{10641625903357416830693572717234701141841485273563233367471} a^{13} + \frac{4446664128721239431150308014066263073233083843346649066252}{10641625903357416830693572717234701141841485273563233367471} a^{12} + \frac{3636001956978878825865388296759135419381653494238269768656}{10641625903357416830693572717234701141841485273563233367471} a^{11} - \frac{3188163070904054870259795193543436507419724780333554866765}{10641625903357416830693572717234701141841485273563233367471} a^{10} - \frac{2838698563072303079595489608337009600502111560266999561184}{10641625903357416830693572717234701141841485273563233367471} a^{9} - \frac{4857537421370133017559464457960560878227089340608624783628}{10641625903357416830693572717234701141841485273563233367471} a^{8} + \frac{671908876851061914044109400762226192386173509871124810087}{10641625903357416830693572717234701141841485273563233367471} a^{7} - \frac{2152206834897268843439742049762946704481279903642564034548}{10641625903357416830693572717234701141841485273563233367471} a^{6} - \frac{563321053251288244463401721256911251978092026992842883275}{10641625903357416830693572717234701141841485273563233367471} a^{5} + \frac{200087102889680447307154732802953475016238235891484884664}{10641625903357416830693572717234701141841485273563233367471} a^{4} + \frac{2584881883192424593459076200374887106792460680120560617179}{10641625903357416830693572717234701141841485273563233367471} a^{3} - \frac{2501835579560285854576979027400843695331526064032477403230}{10641625903357416830693572717234701141841485273563233367471} a^{2} + \frac{1649701848815101718126450762724977278104258731795189581918}{10641625903357416830693572717234701141841485273563233367471} a + \frac{4633474733551086041360407556331464304121627995561202368678}{10641625903357416830693572717234701141841485273563233367471}$

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:  $8$
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:  \( 381001007.919 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n1022
Character table for t16n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.775.1, 8.2.9235210000.1

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
Arithmetically equvalently 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 $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.12.22$x^{8} + 4 x^{7} + 16 x^{3} + 48$$4$$2$$12$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
5Data not computed
$31$31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
31.8.6.3$x^{8} - 31 x^{4} + 11532$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$