↑

Properties

Label 16.0.27510376712...625.44
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{12}$
Root discriminant $106.53$
Ramified primes $5, 101$
Class number $6400$ (GRH)
Class group $[2, 2, 2, 20, 40]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4569760000, -2109120000, 744952000, -255944000, 71416400, -31402800, 9041980, -1893360, 267861, 57043, 17972, -4895, 111, -95, -12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 12*x^14 - 95*x^13 + 111*x^12 - 4895*x^11 + 17972*x^10 + 57043*x^9 + 267861*x^8 - 1893360*x^7 + 9041980*x^6 - 31402800*x^5 + 71416400*x^4 - 255944000*x^3 + 744952000*x^2 - 2109120000*x + 4569760000)
 
gp: K = bnfinit(x^16 - x^15 - 12*x^14 - 95*x^13 + 111*x^12 - 4895*x^11 + 17972*x^10 + 57043*x^9 + 267861*x^8 - 1893360*x^7 + 9041980*x^6 - 31402800*x^5 + 71416400*x^4 - 255944000*x^3 + 744952000*x^2 - 2109120000*x + 4569760000, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 12 x^{14} - 95 x^{13} + 111 x^{12} - 4895 x^{11} + 17972 x^{10} + 57043 x^{9} + 267861 x^{8} - 1893360 x^{7} + 9041980 x^{6} - 31402800 x^{5} + 71416400 x^{4} - 255944000 x^{3} + 744952000 x^{2} - 2109120000 x + 4569760000 \)

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:  \(275103767122062920083301025390625=5^{12}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.53$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{10} - \frac{1}{20} a^{9} + \frac{2}{5} a^{8} + \frac{1}{4} a^{7} - \frac{9}{20} a^{6} + \frac{1}{4} a^{5} - \frac{2}{5} a^{4} + \frac{3}{20} a^{3} + \frac{1}{20} a^{2}$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{10} + \frac{1}{5} a^{9} + \frac{1}{8} a^{8} - \frac{9}{40} a^{7} + \frac{1}{8} a^{6} - \frac{1}{5} a^{5} - \frac{17}{40} a^{4} - \frac{19}{40} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4800} a^{12} - \frac{17}{1600} a^{11} + \frac{19}{2400} a^{10} + \frac{101}{960} a^{9} + \frac{61}{4800} a^{8} - \frac{3}{320} a^{7} + \frac{187}{800} a^{6} + \frac{43}{4800} a^{5} - \frac{363}{1600} a^{4} + \frac{239}{480} a^{3} - \frac{61}{240} a^{2} + \frac{11}{24} a + \frac{1}{12}$, $\frac{1}{25832356651201776000} a^{13} + \frac{1209880681387039}{25832356651201776000} a^{12} - \frac{617104516376264}{201815286337513875} a^{11} + \frac{221968172765369}{90639847898953600} a^{10} + \frac{5984886634485883391}{25832356651201776000} a^{9} + \frac{2307191353155434729}{5166471330240355200} a^{8} - \frac{168179361912102557}{1076348193800074000} a^{7} - \frac{12604816952306315777}{25832356651201776000} a^{6} - \frac{7916970707311069499}{25832356651201776000} a^{5} + \frac{634916044084268113}{1291617832560088800} a^{4} + \frac{1639198588347403}{11329980987369200} a^{3} + \frac{1006688987969957}{4967760894461880} a^{2} + \frac{315787211421547}{8072611453500555} a + \frac{78421667394253}{496776089446188}$, $\frac{1}{671641272931246176000} a^{14} - \frac{1}{671641272931246176000} a^{13} - \frac{3375290291057329}{83955159116405772000} a^{12} - \frac{190222647157689259}{26865650917249847040} a^{11} + \frac{5106596369212193551}{671641272931246176000} a^{10} + \frac{18993495674726718401}{134328254586249235200} a^{9} - \frac{1994577803329462991}{5247197444775360750} a^{8} - \frac{112699460892911156057}{671641272931246176000} a^{7} - \frac{205068916807716320779}{671641272931246176000} a^{6} + \frac{11363929408383551699}{33582063646562308800} a^{5} - \frac{481920795994955387}{2098878977910144300} a^{4} + \frac{781083111139567}{8072611453500555} a^{3} - \frac{303906921725227919}{839551591164057720} a^{2} - \frac{5334894378001001}{12916178325600888} a + \frac{51233533064063}{248388044723094}$, $\frac{1}{87313365481062002880000} a^{15} - \frac{1}{87313365481062002880000} a^{14} - \frac{1}{7276113790088500240000} a^{13} - \frac{228814159102715513}{5820891032070800192000} a^{12} + \frac{157525168495977690311}{87313365481062002880000} a^{11} - \frac{15578994346753284659}{17462673096212400576000} a^{10} + \frac{3976151590970836237943}{21828341370265500720000} a^{9} - \frac{32902616730361113785557}{87313365481062002880000} a^{8} - \frac{13778903697627639053539}{87313365481062002880000} a^{7} - \frac{128493160811119736019}{363805689504425012000} a^{6} + \frac{9153364469593613617}{76590671474615792000} a^{5} - \frac{13905882980303393561}{33582063646562308800} a^{4} + \frac{3987409600960297427}{14552227580177000480} a^{3} + \frac{30763109599297097}{839551591164057720} a^{2} + \frac{28924347276583}{64580891628004440} a - \frac{11096180179613}{248388044723094}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{20}\times C_{40}$, which has order $6400$ (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:  \( -\frac{222774177}{861078555040059200} a^{15} - \frac{23836836939}{11194021215520769600} a^{14} + \frac{222774177}{699626325970048100} a^{13} + \frac{111831229371}{2238804243104153920} a^{12} + \frac{2913217912629}{11194021215520769600} a^{11} + \frac{3342280977531}{2238804243104153920} a^{10} + \frac{9047527650501}{1399252651940096200} a^{9} - \frac{362119201939323}{11194021215520769600} a^{8} - \frac{2730735802299161}{11194021215520769600} a^{7} - \frac{249070663627257}{559701060776038480} a^{6} + \frac{11142496011009}{13992526519400962} a^{5} - \frac{102769960559463}{27985053038801924} a^{4} + \frac{5237420901270}{538174096900037} a^{3} - \frac{9013492918088759}{139925265194009620} a^{2} + \frac{2027245010700}{41398007453849} a + \frac{3764883591300}{41398007453849} \) (order $10$)
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:  \( 6501070.6915 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}, \sqrt{101})\), 4.0.25757525.1 x2, 4.0.5151505.1 x2, \(\Q(\zeta_{5})\), 4.0.1275125.2, 8.0.663450094125625.5, 8.0.1625943765625.6, 8.8.16586252353140625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$