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Properties

Label 16.0.67233141694...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{14}\cdot 29^{6}\cdot 41^{4}$
Root discriminant $73.15$
Ramified primes $2, 5, 29, 41$
Class number $12672$ (GRH)
Class group $[2, 2, 2, 2, 792]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8643461, 10628644, 12306656, 7255146, 4821683, 2001274, 1138799, 362188, 169508, 34754, 16241, 1768, 1033, 18, 49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 49*x^14 + 18*x^13 + 1033*x^12 + 1768*x^11 + 16241*x^10 + 34754*x^9 + 169508*x^8 + 362188*x^7 + 1138799*x^6 + 2001274*x^5 + 4821683*x^4 + 7255146*x^3 + 12306656*x^2 + 10628644*x + 8643461)
 
gp: K = bnfinit(x^16 - 2*x^15 + 49*x^14 + 18*x^13 + 1033*x^12 + 1768*x^11 + 16241*x^10 + 34754*x^9 + 169508*x^8 + 362188*x^7 + 1138799*x^6 + 2001274*x^5 + 4821683*x^4 + 7255146*x^3 + 12306656*x^2 + 10628644*x + 8643461, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 49 x^{14} + 18 x^{13} + 1033 x^{12} + 1768 x^{11} + 16241 x^{10} + 34754 x^{9} + 169508 x^{8} + 362188 x^{7} + 1138799 x^{6} + 2001274 x^{5} + 4821683 x^{4} + 7255146 x^{3} + 12306656 x^{2} + 10628644 x + 8643461 \)

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:  \(672331416948912400000000000000=2^{16}\cdot 5^{14}\cdot 29^{6}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.15$
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:  $2, 5, 29, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} - \frac{1}{2} a^{6} + \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{3}{16} a^{8} - \frac{5}{16} a^{7} - \frac{1}{4} a^{6} - \frac{7}{16} a^{5} + \frac{1}{4} a^{4} - \frac{7}{16} a^{3} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{4348352159374352138011002307434458110875856} a^{15} - \frac{92898202926695896062095101534911362918647}{4348352159374352138011002307434458110875856} a^{14} + \frac{195903891012269057238066763935801954841419}{4348352159374352138011002307434458110875856} a^{13} + \frac{24837673936386176457486252093461862416911}{1087088039843588034502750576858614527718964} a^{12} - \frac{62084413758861506764134120060064288030267}{543544019921794017251375288429307263859482} a^{11} + \frac{6645693971399947330669998063827221847571}{36850442028596204559415273791817441617592} a^{10} + \frac{1099162950689778572070950983916440489982091}{4348352159374352138011002307434458110875856} a^{9} + \frac{33142385051196589812904874702993654341083}{73700884057192409118830547583634883235184} a^{8} + \frac{93676803364959574715987057768874766934955}{543544019921794017251375288429307263859482} a^{7} - \frac{1749514777499822328586073028436661356007543}{4348352159374352138011002307434458110875856} a^{6} + \frac{131087391062712552554258835751453555894262}{271772009960897008625687644214653631929741} a^{5} - \frac{1272630983461071427489606702550630191731375}{4348352159374352138011002307434458110875856} a^{4} - \frac{19694286685116711563916870926687701494935}{271772009960897008625687644214653631929741} a^{3} + \frac{2125397685682956647938143051086306604303801}{4348352159374352138011002307434458110875856} a^{2} + \frac{1802760250992251519361800931619516193155253}{4348352159374352138011002307434458110875856} a - \frac{284170066986306860931979695331076585907067}{2174176079687176069005501153717229055437928}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{792}$, which has order $12672$ (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:  \( 12061.7171318 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.58000.1, \(\Q(\zeta_{20})^+\), 4.4.725.1, 8.8.3364000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed