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Properties

Label 16.0.14461433542...0624.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 13^{6}\cdot 17^{8}$
Root discriminant $43.15$
Ramified primes $2, 13, 17$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $(C_2^2\times D_4).C_2^3$ (as 16T600)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3398, 844, 10470, -4648, 3641, -4440, 5202, -800, 1845, 940, -270, -24, 107, 0, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 10*x^14 + 107*x^12 - 24*x^11 - 270*x^10 + 940*x^9 + 1845*x^8 - 800*x^7 + 5202*x^6 - 4440*x^5 + 3641*x^4 - 4648*x^3 + 10470*x^2 + 844*x + 3398)
 
gp: K = bnfinit(x^16 - 10*x^14 + 107*x^12 - 24*x^11 - 270*x^10 + 940*x^9 + 1845*x^8 - 800*x^7 + 5202*x^6 - 4440*x^5 + 3641*x^4 - 4648*x^3 + 10470*x^2 + 844*x + 3398, 1)
 

Normalized defining polynomial

\( x^{16} - 10 x^{14} + 107 x^{12} - 24 x^{11} - 270 x^{10} + 940 x^{9} + 1845 x^{8} - 800 x^{7} + 5202 x^{6} - 4440 x^{5} + 3641 x^{4} - 4648 x^{3} + 10470 x^{2} + 844 x + 3398 \)

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:  \(144614335422665336893210624=2^{32}\cdot 13^{6}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.15$
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:  $2, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{32} a^{10} - \frac{1}{16} a^{9} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{5}{32} a^{6} + \frac{7}{16} a^{5} + \frac{1}{32} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{3}{16}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{3}{64} a^{9} + \frac{1}{64} a^{8} - \frac{1}{64} a^{7} - \frac{15}{64} a^{6} + \frac{3}{64} a^{5} - \frac{3}{64} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{11}{32} a + \frac{9}{32}$, $\frac{1}{256} a^{12} - \frac{1}{32} a^{9} - \frac{5}{128} a^{8} + \frac{3}{32} a^{7} - \frac{25}{128} a^{6} + \frac{5}{64} a^{5} + \frac{19}{256} a^{4} - \frac{1}{4} a^{3} + \frac{37}{128} a^{2} - \frac{5}{64} a + \frac{55}{128}$, $\frac{1}{512} a^{13} - \frac{1}{512} a^{12} - \frac{1}{64} a^{10} + \frac{15}{256} a^{9} + \frac{1}{256} a^{8} - \frac{5}{256} a^{7} + \frac{3}{256} a^{6} + \frac{159}{512} a^{5} + \frac{13}{512} a^{4} - \frac{91}{256} a^{3} - \frac{79}{256} a^{2} + \frac{97}{256} a - \frac{23}{256}$, $\frac{1}{1024} a^{14} + \frac{1}{1024} a^{12} - \frac{1}{128} a^{11} - \frac{5}{512} a^{10} - \frac{3}{64} a^{9} + \frac{1}{256} a^{8} - \frac{21}{256} a^{7} + \frac{97}{1024} a^{6} + \frac{37}{256} a^{5} - \frac{99}{1024} a^{4} + \frac{11}{256} a^{3} - \frac{57}{128} a^{2} + \frac{91}{256} a - \frac{137}{512}$, $\frac{1}{940450677591279069184} a^{15} + \frac{190756795052556777}{940450677591279069184} a^{14} + \frac{18132100713562161}{940450677591279069184} a^{13} - \frac{966990908480070187}{940450677591279069184} a^{12} + \frac{758527172791550375}{470225338795639534592} a^{11} - \frac{3616692166869436853}{470225338795639534592} a^{10} + \frac{845402826027202253}{235112669397819767296} a^{9} - \frac{1450620301021801731}{117556334698909883648} a^{8} + \frac{7358836402664820173}{940450677591279069184} a^{7} + \frac{214210290629066168981}{940450677591279069184} a^{6} + \frac{298582029413386042705}{940450677591279069184} a^{5} - \frac{12598941678489259187}{940450677591279069184} a^{4} - \frac{69187531393887053175}{235112669397819767296} a^{3} + \frac{444680592974031085}{10222289973818250752} a^{2} + \frac{96317814288614604933}{470225338795639534592} a - \frac{175434498802948437925}{470225338795639534592}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 5108183.48464 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times D_4).C_2^3$ (as 16T600):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 40 conjugacy class representatives for $(C_2^2\times D_4).C_2^3$
Character table for $(C_2^2\times D_4).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.120224.2, 4.0.3757.1, 4.4.9248.1, 8.0.14453810176.7

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.18.8$x^{8} + 36$$4$$2$$18$$Q_8:C_2$$[2, 3, 7/2]^{2}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.3.3$x^{4} + 26$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$