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Properties

Label 16.8.81000012766...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{10}\cdot 149^{4}$
Root discriminant $64.09$
Ramified primes $5, 29, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_4.C_2^2:D_4$ (as 16T305)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6305231, -13009425, 10222736, -3886533, -401729, 1198290, -399834, 97286, 28939, -28831, 4386, 342, -565, 113, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 5*x^14 + 113*x^13 - 565*x^12 + 342*x^11 + 4386*x^10 - 28831*x^9 + 28939*x^8 + 97286*x^7 - 399834*x^6 + 1198290*x^5 - 401729*x^4 - 3886533*x^3 + 10222736*x^2 - 13009425*x + 6305231)
 
gp: K = bnfinit(x^16 - 3*x^15 + 5*x^14 + 113*x^13 - 565*x^12 + 342*x^11 + 4386*x^10 - 28831*x^9 + 28939*x^8 + 97286*x^7 - 399834*x^6 + 1198290*x^5 - 401729*x^4 - 3886533*x^3 + 10222736*x^2 - 13009425*x + 6305231, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 5 x^{14} + 113 x^{13} - 565 x^{12} + 342 x^{11} + 4386 x^{10} - 28831 x^{9} + 28939 x^{8} + 97286 x^{7} - 399834 x^{6} + 1198290 x^{5} - 401729 x^{4} - 3886533 x^{3} + 10222736 x^{2} - 13009425 x + 6305231 \)

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:  \(81000012766225321509609765625=5^{8}\cdot 29^{10}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.09$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{899715865586593632567019090259738562047364619} a^{15} + \frac{59840005323576442930248496984013675029312392}{899715865586593632567019090259738562047364619} a^{14} + \frac{8814383020833608718155921871174065379577784}{128530837940941947509574155751391223149623517} a^{13} + \frac{2419761457585442908702939804655426636298928}{128530837940941947509574155751391223149623517} a^{12} + \frac{345093044685010113254074772272223935916536511}{899715865586593632567019090259738562047364619} a^{11} - \frac{348147415644817628160319544202953785124743577}{899715865586593632567019090259738562047364619} a^{10} - \frac{209552028505274103354193402428263912696619378}{899715865586593632567019090259738562047364619} a^{9} + \frac{156067859557645425762787480495109766767665849}{899715865586593632567019090259738562047364619} a^{8} - \frac{211791142494423585449281359733519582592976223}{899715865586593632567019090259738562047364619} a^{7} + \frac{355803476870566693644583944572441751523074623}{899715865586593632567019090259738562047364619} a^{6} - \frac{342317185554338086461687256586664107845012277}{899715865586593632567019090259738562047364619} a^{5} + \frac{34394219033723040341721987510462828251142136}{128530837940941947509574155751391223149623517} a^{4} - \frac{347481407925156714074681897973508691397785178}{899715865586593632567019090259738562047364619} a^{3} - \frac{219828255336441322470681274490436285177787502}{899715865586593632567019090259738562047364619} a^{2} + \frac{198377662787575998275825132303381440674362427}{899715865586593632567019090259738562047364619} a + \frac{45444802580296620845596167252989971207927177}{899715865586593632567019090259738562047364619}$

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:  \( 107561293.978 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.C_2^2:D_4$ (as 16T305):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 29 conjugacy class representatives for $C_4.C_2^2:D_4$
Character table for $C_4.C_2^2:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), 4.4.725.1 x2, 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
149Data not computed