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Properties

Label 16.8.27752432990...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $3^{8}\cdot 5^{8}\cdot 101^{8}$
Root discriminant $38.92$
Ramified primes $3, 5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2.SD_{16}$ (as 16T163)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1817, 683, 9313, 2633, -2461, -6784, -3679, 1662, 1660, 975, -112, -258, 53, 11, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 16*x^14 + 11*x^13 + 53*x^12 - 258*x^11 - 112*x^10 + 975*x^9 + 1660*x^8 + 1662*x^7 - 3679*x^6 - 6784*x^5 - 2461*x^4 + 2633*x^3 + 9313*x^2 + 683*x - 1817)
 
gp: K = bnfinit(x^16 - x^15 - 16*x^14 + 11*x^13 + 53*x^12 - 258*x^11 - 112*x^10 + 975*x^9 + 1660*x^8 + 1662*x^7 - 3679*x^6 - 6784*x^5 - 2461*x^4 + 2633*x^3 + 9313*x^2 + 683*x - 1817, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 16 x^{14} + 11 x^{13} + 53 x^{12} - 258 x^{11} - 112 x^{10} + 975 x^{9} + 1660 x^{8} + 1662 x^{7} - 3679 x^{6} - 6784 x^{5} - 2461 x^{4} + 2633 x^{3} + 9313 x^{2} + 683 x - 1817 \)

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:  \(27752432990725912250390625=3^{8}\cdot 5^{8}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.92$
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:  $3, 5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{45} a^{12} - \frac{1}{15} a^{11} + \frac{1}{9} a^{10} + \frac{17}{45} a^{9} - \frac{2}{5} a^{8} - \frac{2}{9} a^{7} - \frac{1}{9} a^{6} - \frac{1}{45} a^{5} + \frac{14}{45} a^{4} + \frac{19}{45} a^{3} - \frac{8}{45} a^{2} + \frac{1}{3} a - \frac{8}{45}$, $\frac{1}{45} a^{13} - \frac{4}{45} a^{11} + \frac{2}{45} a^{10} + \frac{2}{5} a^{9} + \frac{11}{45} a^{8} - \frac{4}{9} a^{7} - \frac{16}{45} a^{6} - \frac{19}{45} a^{5} + \frac{16}{45} a^{4} + \frac{19}{45} a^{3} + \frac{2}{15} a^{2} + \frac{22}{45} a - \frac{1}{5}$, $\frac{1}{135} a^{14} + \frac{1}{135} a^{13} + \frac{16}{135} a^{11} - \frac{1}{27} a^{10} - \frac{8}{135} a^{9} + \frac{1}{15} a^{8} - \frac{1}{135} a^{7} + \frac{13}{27} a^{6} + \frac{23}{135} a^{5} + \frac{31}{135} a^{4} - \frac{64}{135} a^{3} - \frac{49}{135} a^{2} + \frac{28}{135} a + \frac{34}{135}$, $\frac{1}{87842235574280868377685615} a^{15} + \frac{39626980913643873486905}{17568447114856173675537123} a^{14} + \frac{318087862239166331231716}{29280745191426956125895205} a^{13} - \frac{32665876939423197136306}{3819227633664385581638505} a^{12} + \frac{2019866250247266166605092}{17568447114856173675537123} a^{11} - \frac{1241761022121391011875824}{17568447114856173675537123} a^{10} + \frac{694738087314538664824616}{1722396775966291536817365} a^{9} - \frac{72979053855707823187574}{7985657779480078943425965} a^{8} - \frac{33372389398352118415476814}{87842235574280868377685615} a^{7} - \frac{12178702007085087779239576}{87842235574280868377685615} a^{6} + \frac{12690112016950716868278646}{87842235574280868377685615} a^{5} - \frac{12690728197892648491303552}{87842235574280868377685615} a^{4} - \frac{5143474441889708729469848}{17568447114856173675537123} a^{3} - \frac{35885384878878620811642197}{87842235574280868377685615} a^{2} - \frac{26788707777257708382002816}{87842235574280868377685615} a - \frac{327750168986947051851137}{1273075877888128527212835}$

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:  \( 5714093.19611 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.SD_{16}$ (as 16T163):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 19 conjugacy class representatives for $C_2^2.SD_{16}$
Character table for $C_2^2.SD_{16}$

Intermediate fields

\(\Q(\sqrt{101}) \), 4.4.51005.1, 8.8.1053611560125.1, 8.4.1053611560125.1, 8.4.65037750625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$