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Properties

Label 16.0.85587369363...3933.1
Degree $16$
Signature $[0, 8]$
Discriminant $11^{10}\cdot 53^{9}$
Root discriminant $41.76$
Ramified primes $11, 53$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^3\times C_4).D_4$ (as 16T675)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12511, -215, 23559, -11799, -4216, -2812, 12815, -14827, 7137, -99, -894, 254, 40, -26, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 26*x^13 + 40*x^12 + 254*x^11 - 894*x^10 - 99*x^9 + 7137*x^8 - 14827*x^7 + 12815*x^6 - 2812*x^5 - 4216*x^4 - 11799*x^3 + 23559*x^2 - 215*x + 12511)
 
gp: K = bnfinit(x^16 - 26*x^13 + 40*x^12 + 254*x^11 - 894*x^10 - 99*x^9 + 7137*x^8 - 14827*x^7 + 12815*x^6 - 2812*x^5 - 4216*x^4 - 11799*x^3 + 23559*x^2 - 215*x + 12511, 1)
 

Normalized defining polynomial

\( x^{16} - 26 x^{13} + 40 x^{12} + 254 x^{11} - 894 x^{10} - 99 x^{9} + 7137 x^{8} - 14827 x^{7} + 12815 x^{6} - 2812 x^{5} - 4216 x^{4} - 11799 x^{3} + 23559 x^{2} - 215 x + 12511 \)

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:  \(85587369363492766398473933=11^{10}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.76$
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:  $11, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{21} a^{11} - \frac{1}{7} a^{10} - \frac{2}{21} a^{9} - \frac{3}{7} a^{8} + \frac{1}{3} a^{7} + \frac{3}{7} a^{5} + \frac{4}{21} a^{4} - \frac{1}{3} a^{3} + \frac{2}{21} a^{2} - \frac{5}{21}$, $\frac{1}{21} a^{12} + \frac{1}{7} a^{10} - \frac{1}{21} a^{9} + \frac{1}{21} a^{8} + \frac{2}{21} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{21} a^{3} - \frac{1}{21} a^{2} + \frac{3}{7} a - \frac{1}{21}$, $\frac{1}{21} a^{13} + \frac{1}{21} a^{10} + \frac{2}{7} a^{8} + \frac{2}{21} a^{7} - \frac{4}{21} a^{6} - \frac{1}{21} a^{5} - \frac{1}{7} a^{4} - \frac{1}{21} a^{3} - \frac{4}{21} a^{2} - \frac{8}{21} a + \frac{8}{21}$, $\frac{1}{705285} a^{14} + \frac{61}{6717} a^{13} + \frac{16193}{705285} a^{12} - \frac{5036}{705285} a^{11} - \frac{5071}{705285} a^{10} - \frac{102839}{705285} a^{9} + \frac{4198}{705285} a^{8} + \frac{320414}{705285} a^{7} - \frac{200239}{705285} a^{6} + \frac{18713}{47019} a^{5} + \frac{2166}{11195} a^{4} + \frac{19919}{100755} a^{3} + \frac{123098}{705285} a^{2} - \frac{48796}{141057} a + \frac{280123}{705285}$, $\frac{1}{8863440207541171970194310715} a^{15} - \frac{1078975321310828083459}{1772688041508234394038862143} a^{14} + \frac{30141270384915214359288548}{8863440207541171970194310715} a^{13} - \frac{3384460401241329038483784}{984826689726796885577145635} a^{12} + \frac{66095454821870980804139918}{2954480069180390656731436905} a^{11} - \frac{183328544718451827508890548}{2954480069180390656731436905} a^{10} - \frac{477738448391623113320803117}{8863440207541171970194310715} a^{9} - \frac{885957282687278623488185117}{2954480069180390656731436905} a^{8} - \frac{761141704848167812457683994}{8863440207541171970194310715} a^{7} + \frac{423013300574119886257112134}{1772688041508234394038862143} a^{6} + \frac{393554351495086185494255926}{2954480069180390656731436905} a^{5} + \frac{1110574684617201348863957018}{8863440207541171970194310715} a^{4} - \frac{3077221166380268293312445432}{8863440207541171970194310715} a^{3} + \frac{34053979218219659661483568}{590896013836078131346287381} a^{2} - \frac{2085773713840520527903716382}{8863440207541171970194310715} a + \frac{377595518383829122670321300}{1772688041508234394038862143}$

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:  $7$
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:  \( 4006604.00833 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^3\times C_4).D_4$ (as 16T675):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$
Character table for $(C_2^3\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.6413.1, 8.0.2179708157.1

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 $16$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$11$11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$53$53.4.3.3$x^{4} + 106$$4$$1$$3$$C_4$$[\ ]_{4}$
53.4.0.1$x^{4} - x + 18$$1$$4$$0$$C_4$$[\ ]^{4}$
53.8.6.2$x^{8} + 477 x^{4} + 70225$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$