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Properties

Label 16.8.17690457232...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{14}\cdot 11^{12}\cdot 31^{4}$
Root discriminant $58.28$
Ramified primes $5, 11, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-668819, 1482888, -305141, -1607112, 1428728, -55712, -509054, 281034, -28502, -29212, 12624, -996, -667, 206, -9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 9*x^14 + 206*x^13 - 667*x^12 - 996*x^11 + 12624*x^10 - 29212*x^9 - 28502*x^8 + 281034*x^7 - 509054*x^6 - 55712*x^5 + 1428728*x^4 - 1607112*x^3 - 305141*x^2 + 1482888*x - 668819)
 
gp: K = bnfinit(x^16 - 6*x^15 - 9*x^14 + 206*x^13 - 667*x^12 - 996*x^11 + 12624*x^10 - 29212*x^9 - 28502*x^8 + 281034*x^7 - 509054*x^6 - 55712*x^5 + 1428728*x^4 - 1607112*x^3 - 305141*x^2 + 1482888*x - 668819, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 9 x^{14} + 206 x^{13} - 667 x^{12} - 996 x^{11} + 12624 x^{10} - 29212 x^{9} - 28502 x^{8} + 281034 x^{7} - 509054 x^{6} - 55712 x^{5} + 1428728 x^{4} - 1607112 x^{3} - 305141 x^{2} + 1482888 x - 668819 \)

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:  \(17690457232041959478759765625=5^{14}\cdot 11^{12}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.28$
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:  $5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{38} a^{14} + \frac{9}{38} a^{13} - \frac{1}{38} a^{12} - \frac{1}{19} a^{11} - \frac{1}{38} a^{9} - \frac{7}{38} a^{8} - \frac{3}{19} a^{7} - \frac{1}{38} a^{6} - \frac{4}{19} a^{5} - \frac{9}{19} a^{4} - \frac{9}{19} a^{3} - \frac{13}{38} a^{2} - \frac{15}{38} a$, $\frac{1}{8578270783112503839582950207891618702} a^{15} - \frac{8366708822501297918436720899517182}{4289135391556251919791475103945809351} a^{14} + \frac{757677307378748586628062666271486647}{4289135391556251919791475103945809351} a^{13} - \frac{1981407087314545567620563796625542097}{8578270783112503839582950207891618702} a^{12} - \frac{1875084159802471811805461083145131275}{8578270783112503839582950207891618702} a^{11} - \frac{1604929848056128026508285558190900465}{8578270783112503839582950207891618702} a^{10} - \frac{294511968499664935873010361680513184}{4289135391556251919791475103945809351} a^{9} - \frac{968031612301401855036041885513261947}{8578270783112503839582950207891618702} a^{8} + \frac{1309784824910752829074841400153886972}{4289135391556251919791475103945809351} a^{7} - \frac{1686484509971029988703833517269542458}{4289135391556251919791475103945809351} a^{6} - \frac{1735788161423694144215686828556171417}{4289135391556251919791475103945809351} a^{5} - \frac{3026400184164996195838550772255502177}{8578270783112503839582950207891618702} a^{4} + \frac{1654069375026021706575000522309848165}{8578270783112503839582950207891618702} a^{3} + \frac{2231184046275314366861898064808364025}{8578270783112503839582950207891618702} a^{2} + \frac{1217720521803964376703604411080205981}{4289135391556251919791475103945809351} a + \frac{68410786946259121649520552363995137}{451487935953289675767523695152190458}$

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:  $11$
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:  \( 104019145.481 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.15125.1, 8.4.78009078125.1

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
31Data not computed