↑

Properties

Label 16.8.41587707333...1009.1
Degree $16$
Signature $[8, 4]$
Discriminant $17^{14}\cdot 89^{12}$
Root discriminant $345.69$
Ramified primes $17, 89$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-804002796719, 1002903417071, -491053716521, 14877745689, 54011941451, -12816699466, 1709408121, -504128958, 71363414, -16878, -455761, 78999, -8849, 89, -49, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 49*x^14 + 89*x^13 - 8849*x^12 + 78999*x^11 - 455761*x^10 - 16878*x^9 + 71363414*x^8 - 504128958*x^7 + 1709408121*x^6 - 12816699466*x^5 + 54011941451*x^4 + 14877745689*x^3 - 491053716521*x^2 + 1002903417071*x - 804002796719)
 
gp: K = bnfinit(x^16 - 2*x^15 - 49*x^14 + 89*x^13 - 8849*x^12 + 78999*x^11 - 455761*x^10 - 16878*x^9 + 71363414*x^8 - 504128958*x^7 + 1709408121*x^6 - 12816699466*x^5 + 54011941451*x^4 + 14877745689*x^3 - 491053716521*x^2 + 1002903417071*x - 804002796719, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 49 x^{14} + 89 x^{13} - 8849 x^{12} + 78999 x^{11} - 455761 x^{10} - 16878 x^{9} + 71363414 x^{8} - 504128958 x^{7} + 1709408121 x^{6} - 12816699466 x^{5} + 54011941451 x^{4} + 14877745689 x^{3} - 491053716521 x^{2} + 1002903417071 x - 804002796719 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(41587707333348149512906399539663989371009=17^{14}\cdot 89^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $345.69$
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:  $17, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{1004} a^{14} - \frac{3}{502} a^{13} - \frac{35}{251} a^{12} - \frac{167}{1004} a^{11} - \frac{113}{1004} a^{10} - \frac{119}{502} a^{9} + \frac{112}{251} a^{8} + \frac{83}{502} a^{7} + \frac{61}{502} a^{6} - \frac{239}{502} a^{5} + \frac{161}{1004} a^{4} - \frac{155}{502} a^{3} - \frac{137}{502} a^{2} - \frac{99}{1004} a + \frac{67}{1004}$, $\frac{1}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208} a^{15} - \frac{190325011271683375818662478609619858458679094528328529395518484300112027785104863}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208} a^{14} - \frac{30519345327190807659340590443882206467349642278683975093519625271928212815291452387}{471900878851046355073507657619651769731073797678565464212855685202294760082247084604} a^{13} + \frac{139093826352320469194822915815197947978709093171937310187431517240777029640094742091}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208} a^{12} + \frac{21477723237759908659959678932956815492041026609262369956247360532355869150347797028}{117975219712761588768376914404912942432768449419641366053213921300573690020561771151} a^{11} - \frac{99748080225604759332792377078835462859245775992484340420690380088394732362738988081}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208} a^{10} + \frac{1590506140475100221316765603585585559658167975317475257979906965658433885091713602}{117975219712761588768376914404912942432768449419641366053213921300573690020561771151} a^{9} + \frac{3312082131203608376924685813427887972861565946928368929961345069879612330167699647}{7043296699269348583186681457009727906433937278784559167356055003019324777346971412} a^{8} - \frac{16623340201557853217434000898974172545925079869349014704091566443511784302974980769}{117975219712761588768376914404912942432768449419641366053213921300573690020561771151} a^{7} - \frac{81262048951059264312460262736473870290520329459831027506598188924329122050610739991}{471900878851046355073507657619651769731073797678565464212855685202294760082247084604} a^{6} + \frac{126888685152244899990864544353233927944387942094507717951803135212837197509627480115}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208} a^{5} + \frac{420759372358276146213167842774742081960522782501212249971467791432627166092069742575}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208} a^{4} - \frac{52261188558429733895051682501411455906756546928817995114656520869756663325312141205}{117975219712761588768376914404912942432768449419641366053213921300573690020561771151} a^{3} + \frac{110499691772405950295534926632954052045109439006290261418377089065235536343223239721}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208} a^{2} + \frac{139409645689646684644553920864886537264652318856586269710804702580317670766051134789}{471900878851046355073507657619651769731073797678565464212855685202294760082247084604} a - \frac{236255881817944806547204579402041975335328165493520598090525220285579439107328376455}{943801757702092710147015315239303539462147595357130928425711370404589520164494169208}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $11$
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:  \( 59725155189600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

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 $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{1513}) \), \(\Q(\sqrt{89}) \), 4.4.4913.1, 4.4.38915873.1, \(\Q(\sqrt{17}, \sqrt{89})\), 8.8.1514445171352129.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
89Data not computed