↑

Properties

Label 16.0.13279393802...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{15}\cdot 61^{11}$
Root discriminant $76.33$
Ramified primes $5, 61$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group 16T1192

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![195517031, 459900849, 402284320, 130373870, -24060035, -28769888, -4007092, 2393835, 745310, -80835, -61257, -2833, 3085, 170, -75, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 75*x^14 + 170*x^13 + 3085*x^12 - 2833*x^11 - 61257*x^10 - 80835*x^9 + 745310*x^8 + 2393835*x^7 - 4007092*x^6 - 28769888*x^5 - 24060035*x^4 + 130373870*x^3 + 402284320*x^2 + 459900849*x + 195517031)
 
gp: K = bnfinit(x^16 - x^15 - 75*x^14 + 170*x^13 + 3085*x^12 - 2833*x^11 - 61257*x^10 - 80835*x^9 + 745310*x^8 + 2393835*x^7 - 4007092*x^6 - 28769888*x^5 - 24060035*x^4 + 130373870*x^3 + 402284320*x^2 + 459900849*x + 195517031, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 75 x^{14} + 170 x^{13} + 3085 x^{12} - 2833 x^{11} - 61257 x^{10} - 80835 x^{9} + 745310 x^{8} + 2393835 x^{7} - 4007092 x^{6} - 28769888 x^{5} - 24060035 x^{4} + 130373870 x^{3} + 402284320 x^{2} + 459900849 x + 195517031 \)

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:  \(1327939380231806599761962890625=5^{15}\cdot 61^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.33$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{61} a^{14} - \frac{12}{61} a^{13} - \frac{19}{61} a^{12} + \frac{10}{61} a^{11} + \frac{27}{61} a^{10} + \frac{14}{61} a^{9} - \frac{23}{61} a^{8} - \frac{28}{61} a^{7} - \frac{6}{61} a^{6} + \frac{10}{61} a^{5} - \frac{22}{61} a^{4} - \frac{24}{61} a^{2} - \frac{7}{61} a + \frac{5}{61}$, $\frac{1}{12541753342496974724653852687088501784009143754569966748809} a^{15} + \frac{6527037852266213259967667386227873713021859435115063773}{1140159394772452247695804789735318344000831250415451522619} a^{14} + \frac{2279517589523133336371322087745165511646700021232237070915}{12541753342496974724653852687088501784009143754569966748809} a^{13} + \frac{2822143086021121558997343054223519650746583098968553985960}{12541753342496974724653852687088501784009143754569966748809} a^{12} - \frac{5776131695792557133786082554777739332389377301285139823197}{12541753342496974724653852687088501784009143754569966748809} a^{11} - \frac{5343964164205137673755258855638144011970620038860511305209}{12541753342496974724653852687088501784009143754569966748809} a^{10} + \frac{5972950561019342195860036871633230259777615468930716400816}{12541753342496974724653852687088501784009143754569966748809} a^{9} - \frac{5704923996910739963485678457793475869301365412291741417738}{12541753342496974724653852687088501784009143754569966748809} a^{8} + \frac{5787718501964385807784715213620947772013405582664684361767}{12541753342496974724653852687088501784009143754569966748809} a^{7} - \frac{829881708487927069235203862647252210089547720574259836553}{12541753342496974724653852687088501784009143754569966748809} a^{6} - \frac{4812859159205068993936768566757054282869951071086749101317}{12541753342496974724653852687088501784009143754569966748809} a^{5} + \frac{5629851108480627016179496668159627788858126055888396235133}{12541753342496974724653852687088501784009143754569966748809} a^{4} - \frac{3139694122915281997190662626742814613109273373036283121716}{12541753342496974724653852687088501784009143754569966748809} a^{3} + \frac{5340059880492497851893138509224856864721361247325660225631}{12541753342496974724653852687088501784009143754569966748809} a^{2} + \frac{5135673220536763009098982463290124797500066037498141307740}{12541753342496974724653852687088501784009143754569966748809} a + \frac{90190899207445308166357200478430662847590996258833073101}{404572688467644345956575893131887154322875604986127959639}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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{2805456434864313088555702133143942817395326859607}{609385031946794360072584067202201145911721673124239189} a^{15} + \frac{8008958134090900096292018142875268295611826424865}{609385031946794360072584067202201145911721673124239189} a^{14} + \frac{195481385703214171279127657272530266331438045309631}{609385031946794360072584067202201145911721673124239189} a^{13} - \frac{839164133436590690528663110641049973879458176234099}{609385031946794360072584067202201145911721673124239189} a^{12} - \frac{7093938739573112741642497423011914217236556341822847}{609385031946794360072584067202201145911721673124239189} a^{11} + \frac{21075390860589243648382664013074278067918858261277961}{609385031946794360072584067202201145911721673124239189} a^{10} + \frac{132636484240980655688469114753469232039174944250253426}{609385031946794360072584067202201145911721673124239189} a^{9} - \frac{18435558848648835339515176934334832251899253194899921}{609385031946794360072584067202201145911721673124239189} a^{8} - \frac{2054914819850335951060836730390963667725582139176838444}{609385031946794360072584067202201145911721673124239189} a^{7} - \frac{2909185658527727434361022473853006574558393899882784892}{609385031946794360072584067202201145911721673124239189} a^{6} + \frac{16595313065066066613201482614158425397911588004896081265}{609385031946794360072584067202201145911721673124239189} a^{5} + \frac{49948909775071587552444814588072208324516910928960950802}{609385031946794360072584067202201145911721673124239189} a^{4} - \frac{24738920301950089687680089989240533975461967059357486722}{609385031946794360072584067202201145911721673124239189} a^{3} - \frac{319464447447749167183393568641957889628327549456550552554}{609385031946794360072584067202201145911721673124239189} a^{2} - \frac{537743415677093893430392815552437882207522502326842351873}{609385031946794360072584067202201145911721673124239189} a - \frac{9590531594224303270176913343454423091452306001389289205}{19657581675703043873309163458135520835861989455620619} \) (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:  \( 31039360.1909 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1192:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 88 conjugacy class representatives for t16n1192 are not computed
Character table for t16n1192 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.17732890625.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $16$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
5Data not computed
61Data not computed