↑

Properties

Label 16.0.52516204899...2401.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 61^{4}$
Root discriminant $72.03$
Ramified primes $41, 61$
Class number $41$ (GRH)
Class group $[41]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1871824, -1240424, 912840, 202432, -424715, 401269, -136633, -16818, 58718, -35603, 11189, -1282, -400, 160, -8, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824)
 
gp: K = bnfinit(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 400*x^12 - 1282*x^11 + 11189*x^10 - 35603*x^9 + 58718*x^8 - 16818*x^7 - 136633*x^6 + 401269*x^5 - 424715*x^4 + 202432*x^3 + 912840*x^2 - 1240424*x + 1871824, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 8 x^{14} + 160 x^{13} - 400 x^{12} - 1282 x^{11} + 11189 x^{10} - 35603 x^{9} + 58718 x^{8} - 16818 x^{7} - 136633 x^{6} + 401269 x^{5} - 424715 x^{4} + 202432 x^{3} + 912840 x^{2} - 1240424 x + 1871824 \)

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:  \(525162048993676835600871662401=41^{14}\cdot 61^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.03$
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:  $41, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{187912} a^{14} + \frac{15913}{187912} a^{13} - \frac{18591}{187912} a^{12} + \frac{35157}{187912} a^{11} - \frac{25087}{187912} a^{10} - \frac{47149}{187912} a^{9} + \frac{8471}{23489} a^{8} + \frac{26383}{187912} a^{7} - \frac{24269}{187912} a^{6} + \frac{88209}{187912} a^{5} + \frac{1108}{23489} a^{4} - \frac{1901}{187912} a^{3} - \frac{24213}{93956} a^{2} - \frac{1051}{23489} a + \frac{2661}{23489}$, $\frac{1}{488708838952713221947905651093476635768} a^{15} + \frac{52551488926332635439465910271453}{122177209738178305486976412773369158942} a^{14} + \frac{3197954393116789556531399523651035881}{244354419476356610973952825546738317884} a^{13} + \frac{21579514758119188571512308139311612867}{244354419476356610973952825546738317884} a^{12} - \frac{14167802438737455398423800468591775975}{244354419476356610973952825546738317884} a^{11} - \frac{7045815865941621677563697951489094053}{122177209738178305486976412773369158942} a^{10} - \frac{81245336897723447788720650727191163349}{488708838952713221947905651093476635768} a^{9} + \frac{237328044216924088957482597025938979737}{488708838952713221947905651093476635768} a^{8} + \frac{50911840457213547865826877268340812733}{122177209738178305486976412773369158942} a^{7} - \frac{24379437087737200664356989970650669197}{122177209738178305486976412773369158942} a^{6} - \frac{215636772790873109765309002621178561819}{488708838952713221947905651093476635768} a^{5} - \frac{144631032311892750402283375233715413987}{488708838952713221947905651093476635768} a^{4} + \frac{191096995483358708837841896734212075159}{488708838952713221947905651093476635768} a^{3} + \frac{15822595946986194144903547233064309569}{61088604869089152743488206386684579471} a^{2} - \frac{21557484595581842169999146628224671721}{61088604869089152743488206386684579471} a - \frac{25314792752101033990843225674845100444}{61088604869089152743488206386684579471}$

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

Class group and class number

$C_{41}$, which has order $41$ (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:  \( 2911738.86032 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 4.4.102541.1, 4.4.4204181.1, 8.0.724680653111201.1, 8.0.194754273881.1, 8.8.17675137880761.1

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 16 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
41Data not computed
$61$61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
61.8.4.1$x^{8} + 14884 x^{4} - 226981 x^{2} + 55383364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$