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Properties

Label 16.0.81862996112...1809.6
Degree $16$
Signature $[0, 8]$
Discriminant $17^{10}\cdot 67^{8}$
Root discriminant $48.09$
Ramified primes $17, 67$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![77291, 77319, 79362, 81624, 52958, 14085, 4011, 1721, -630, -589, 209, 210, 54, -1, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291)
 
gp: K = bnfinit(x^16 - 8*x^14 - x^13 + 54*x^12 + 210*x^11 + 209*x^10 - 589*x^9 - 630*x^8 + 1721*x^7 + 4011*x^6 + 14085*x^5 + 52958*x^4 + 81624*x^3 + 79362*x^2 + 77319*x + 77291, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - x^{13} + 54 x^{12} + 210 x^{11} + 209 x^{10} - 589 x^{9} - 630 x^{8} + 1721 x^{7} + 4011 x^{6} + 14085 x^{5} + 52958 x^{4} + 81624 x^{3} + 79362 x^{2} + 77319 x + 77291 \)

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:  \(818629961123679547712831809=17^{10}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.09$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{76} a^{13} - \frac{9}{76} a^{12} + \frac{7}{38} a^{11} - \frac{13}{76} a^{10} + \frac{1}{76} a^{9} - \frac{25}{76} a^{8} - \frac{9}{38} a^{7} + \frac{1}{19} a^{6} - \frac{7}{19} a^{5} + \frac{7}{19} a^{4} - \frac{1}{4} a^{3} + \frac{9}{19} a^{2} - \frac{25}{76} a + \frac{5}{19}$, $\frac{1}{4078692} a^{14} + \frac{6167}{2039346} a^{13} - \frac{27361}{226594} a^{12} - \frac{47281}{679782} a^{11} + \frac{151895}{679782} a^{10} - \frac{179125}{453188} a^{9} - \frac{399925}{1019673} a^{8} - \frac{1038125}{4078692} a^{7} - \frac{273413}{679782} a^{6} + \frac{179011}{679782} a^{5} + \frac{220963}{453188} a^{4} + \frac{96665}{1359564} a^{3} - \frac{45313}{1019673} a^{2} - \frac{258205}{4078692} a + \frac{1259731}{4078692}$, $\frac{1}{1350555188693136534485118036} a^{15} - \frac{45705665181745009579}{450185062897712178161706012} a^{14} + \frac{524726821980975502983167}{1350555188693136534485118036} a^{13} + \frac{4836773601580539629514091}{112546265724428044540426503} a^{12} - \frac{18252075848781294954699}{150061687632570726053902004} a^{11} - \frac{19189210304645358424681564}{112546265724428044540426503} a^{10} + \frac{667534121791284126599064401}{1350555188693136534485118036} a^{9} - \frac{296923827511440057358999477}{1350555188693136534485118036} a^{8} + \frac{6072500652674390769983333}{337638797173284133621279509} a^{7} - \frac{12108052526165787325006214}{37515421908142681513475501} a^{6} + \frac{161008833506561615768136013}{450185062897712178161706012} a^{5} - \frac{84031597693344424785839633}{225092531448856089080853006} a^{4} - \frac{57648478481267790327102382}{337638797173284133621279509} a^{3} + \frac{13552047206560897685990407}{112546265724428044540426503} a^{2} + \frac{514070505988909106038184009}{1350555188693136534485118036} a - \frac{113845603640640012941471014}{337638797173284133621279509}$

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

Class group and class number

$C_{4}$, 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:  \( 2782055.44714 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{-1139}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-67}) \), 4.0.76313.1 x2, \(\Q(\sqrt{17}, \sqrt{-67})\), 4.2.19363.1 x2, 8.0.1683041777041.1 x2, 8.0.28611710209697.2 x2, 8.0.1683041777041.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$