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Properties

Label 16.8.10787325443...625.13
Degree $16$
Signature $[8, 4]$
Discriminant $5^{10}\cdot 101^{10}$
Root discriminant $48.93$
Ramified primes $5, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2\times OD_{16}).C_2$ (as 16T123)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2845, 5340, -3956, -7155, -10198, -8586, 16544, 12084, -5348, -1839, -647, -766, 56, 76, -20, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 20*x^14 + 76*x^13 + 56*x^12 - 766*x^11 - 647*x^10 - 1839*x^9 - 5348*x^8 + 12084*x^7 + 16544*x^6 - 8586*x^5 - 10198*x^4 - 7155*x^3 - 3956*x^2 + 5340*x + 2845)
 
gp: K = bnfinit(x^16 - 2*x^15 - 20*x^14 + 76*x^13 + 56*x^12 - 766*x^11 - 647*x^10 - 1839*x^9 - 5348*x^8 + 12084*x^7 + 16544*x^6 - 8586*x^5 - 10198*x^4 - 7155*x^3 - 3956*x^2 + 5340*x + 2845, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 20 x^{14} + 76 x^{13} + 56 x^{12} - 766 x^{11} - 647 x^{10} - 1839 x^{9} - 5348 x^{8} + 12084 x^{7} + 16544 x^{6} - 8586 x^{5} - 10198 x^{4} - 7155 x^{3} - 3956 x^{2} + 5340 x + 2845 \)

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:  \(1078732544346879404306640625=5^{10}\cdot 101^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.93$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{1775} a^{14} + \frac{8}{355} a^{13} + \frac{43}{1775} a^{12} - \frac{673}{1775} a^{11} - \frac{116}{1775} a^{10} - \frac{147}{1775} a^{9} - \frac{299}{1775} a^{8} - \frac{348}{1775} a^{7} + \frac{244}{1775} a^{6} - \frac{702}{1775} a^{5} + \frac{762}{1775} a^{4} - \frac{523}{1775} a^{3} - \frac{518}{1775} a^{2} + \frac{56}{355} a + \frac{102}{355}$, $\frac{1}{82351341628805124363016282110925} a^{15} + \frac{16091156040705622056017988333}{82351341628805124363016282110925} a^{14} + \frac{299644466445377638086946385613}{82351341628805124363016282110925} a^{13} - \frac{7734935145484915480483071593869}{82351341628805124363016282110925} a^{12} - \frac{6229178474217638868747034994759}{16470268325761024872603256422185} a^{11} - \frac{3874785987226416519840378698892}{16470268325761024872603256422185} a^{10} + \frac{1048079391851722771053020342064}{16470268325761024872603256422185} a^{9} - \frac{5706276292828529281544654729861}{16470268325761024872603256422185} a^{8} + \frac{6042653243554953682125897057803}{16470268325761024872603256422185} a^{7} - \frac{1687504582680067656451716386667}{16470268325761024872603256422185} a^{6} - \frac{18006286985149679769918974029409}{82351341628805124363016282110925} a^{5} - \frac{36398842733550367791490480615632}{82351341628805124363016282110925} a^{4} + \frac{24414032151104364054431033682903}{82351341628805124363016282110925} a^{3} - \frac{37506522554081130432126835484114}{82351341628805124363016282110925} a^{2} + \frac{432638652593359027660607890570}{3294053665152204974520651284437} a - \frac{3903287089927665681974969110094}{16470268325761024872603256422185}$

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:  \( 75259408.6384 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).C_2$ (as 16T123):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times OD_{16}).C_2$
Character table for $(C_2\times OD_{16}).C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{505}) \), 4.4.2525.1 x2, 4.4.51005.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.8.65037750625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
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