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Properties

Label 16.0.80073733711...8449.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 73^{14}$
Root discriminant $73.96$
Ramified primes $3, 73$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16112196, -14077098, 12479679, -7629345, 4270839, -1935990, 826306, -304134, 122068, -30644, 12064, -2306, 850, -104, 37, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196)
 
gp: K = bnfinit(x^16 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 37 x^{14} - 104 x^{13} + 850 x^{12} - 2306 x^{11} + 12064 x^{10} - 30644 x^{9} + 122068 x^{8} - 304134 x^{7} + 826306 x^{6} - 1935990 x^{5} + 4270839 x^{4} - 7629345 x^{3} + 12479679 x^{2} - 14077098 x + 16112196 \)

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:  \(800737337114777368288115578449=3^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.96$
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:  $3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{8} + \frac{1}{18} a^{7} - \frac{1}{18} a^{6} + \frac{2}{9} a^{5} - \frac{1}{18} a^{4} - \frac{5}{18} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3} a$, $\frac{1}{36} a^{9} + \frac{1}{6} a^{5} - \frac{13}{36} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{36} a^{10} - \frac{1}{36} a^{4}$, $\frac{1}{144} a^{11} + \frac{1}{144} a^{8} - \frac{5}{72} a^{7} - \frac{1}{18} a^{6} - \frac{5}{144} a^{5} + \frac{7}{36} a^{4} - \frac{1}{36} a^{3} - \frac{5}{16} a^{2} - \frac{11}{24} a - \frac{1}{4}$, $\frac{1}{432} a^{12} + \frac{1}{108} a^{10} + \frac{5}{432} a^{9} - \frac{1}{216} a^{8} - \frac{1}{27} a^{7} - \frac{7}{144} a^{6} - \frac{13}{108} a^{5} - \frac{2}{9} a^{4} + \frac{5}{144} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{432} a^{13} + \frac{1}{432} a^{11} + \frac{5}{432} a^{10} - \frac{1}{216} a^{9} + \frac{5}{432} a^{8} - \frac{5}{144} a^{7} - \frac{1}{108} a^{6} + \frac{13}{144} a^{5} - \frac{5}{48} a^{4} + \frac{31}{72} a^{3} - \frac{7}{16} a^{2} - \frac{5}{24} a + \frac{1}{4}$, $\frac{1}{1093824} a^{14} - \frac{161}{546912} a^{13} + \frac{497}{546912} a^{12} + \frac{71}{34182} a^{11} + \frac{167}{273456} a^{10} - \frac{4657}{546912} a^{9} + \frac{395}{546912} a^{8} - \frac{2933}{136728} a^{7} + \frac{16841}{546912} a^{6} - \frac{11099}{45576} a^{5} + \frac{4397}{91152} a^{4} - \frac{18875}{60768} a^{3} - \frac{42071}{121536} a^{2} - \frac{1157}{20256} a - \frac{323}{3376}$, $\frac{1}{190769474137337809202862485568} a^{15} + \frac{29845484689583844250157}{95384737068668904601431242784} a^{14} - \frac{60912387597379289411872385}{95384737068668904601431242784} a^{13} - \frac{6014109835523700687157}{14399869726550257337172591} a^{12} - \frac{29959369339240108008566707}{23846184267167226150357810696} a^{11} - \frac{1061676511874589045563315231}{95384737068668904601431242784} a^{10} - \frac{410225918070146880057544355}{31794912356222968200477080928} a^{9} - \frac{187568420299616593247462177}{47692368534334452300715621392} a^{8} + \frac{2837213433630256014110421259}{95384737068668904601431242784} a^{7} + \frac{1840010569604071371360690739}{23846184267167226150357810696} a^{6} + \frac{40935111340059603544624543}{3974364044527871025059635116} a^{5} + \frac{1167495014535987341581218041}{31794912356222968200477080928} a^{4} + \frac{445209249397311872777598811}{7065536079160659600106017984} a^{3} - \frac{2789319058063117907808771023}{10598304118740989400159026976} a^{2} + \frac{348329680073406094416423895}{1766384019790164900026504496} a + \frac{55498858707175882131668}{165020928605209725338799}$

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:  \( \frac{14499826137671475455}{226030182627177499055524272} a^{15} - \frac{15987868879075745887}{56507545656794374763881068} a^{14} + \frac{680348569009555995127}{226030182627177499055524272} a^{13} - \frac{1286431468471518157}{121325916600739398312144} a^{12} + \frac{16660316977632515312825}{226030182627177499055524272} a^{11} - \frac{53291839986934039605113}{226030182627177499055524272} a^{10} + \frac{88147140135779286742243}{75343394209059166351841424} a^{9} - \frac{734145426492639988891597}{226030182627177499055524272} a^{8} + \frac{2795745413489208422453131}{226030182627177499055524272} a^{7} - \frac{7023986935344381549435293}{226030182627177499055524272} a^{6} + \frac{6919529768917670372069153}{75343394209059166351841424} a^{5} - \frac{15168196515807677637694225}{75343394209059166351841424} a^{4} + \frac{932710014502687642912481}{2092872061362754620884484} a^{3} - \frac{17501585202279771235073819}{25114464736353055450613808} a^{2} + \frac{5497288394930086996098583}{4185744122725509241768968} a - \frac{1614075467195279358845}{3128358836117719911636} \) (order $6$)
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:  \( 20892766922.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$SD_{16}$ (as 16T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{-219}) \), \(\Q(\sqrt{-3}, \sqrt{73})\), 4.2.1167051.1 x2, 4.0.3501153.2 x2, 8.0.12258072329409.3, 8.2.298279760015619.1 x4

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$