Properties

Label 16.0.685...361.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.856\times 10^{44}$
Root discriminant $634.24$
Ramified primes $13, 89$
Class number $2856068$ (GRH)
Class group $[2856068]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 70*x^14 - 2149*x^13 + 14931*x^12 - 286964*x^11 + 3931708*x^10 - 35756396*x^9 + 326985485*x^8 - 2509676741*x^7 + 16373741504*x^6 - 80620586906*x^5 + 305886025651*x^4 - 925536442193*x^3 + 2156195358075*x^2 - 3016380401225*x + 4337589140875)
 
gp: K = bnfinit(x^16 - x^15 + 70*x^14 - 2149*x^13 + 14931*x^12 - 286964*x^11 + 3931708*x^10 - 35756396*x^9 + 326985485*x^8 - 2509676741*x^7 + 16373741504*x^6 - 80620586906*x^5 + 305886025651*x^4 - 925536442193*x^3 + 2156195358075*x^2 - 3016380401225*x + 4337589140875, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4337589140875, -3016380401225, 2156195358075, -925536442193, 305886025651, -80620586906, 16373741504, -2509676741, 326985485, -35756396, 3931708, -286964, 14931, -2149, 70, -1, 1]);
 

\( x^{16} - x^{15} + 70 x^{14} - 2149 x^{13} + 14931 x^{12} - 286964 x^{11} + 3931708 x^{10} - 35756396 x^{9} + 326985485 x^{8} - 2509676741 x^{7} + 16373741504 x^{6} - 80620586906 x^{5} + 305886025651 x^{4} - 925536442193 x^{3} + 2156195358075 x^{2} - 3016380401225 x + 4337589140875 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(685578251337344185848967768724165145970351361\)\(\medspace = 13^{14}\cdot 89^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $634.24$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 89$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{25} a^{14} + \frac{1}{25} a^{11} + \frac{2}{25} a^{10} - \frac{2}{25} a^{9} + \frac{6}{25} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{6}{25} a^{5} + \frac{3}{25} a^{4} + \frac{12}{25} a^{3} - \frac{12}{25} a^{2} + \frac{2}{5} a$, $\frac{1}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{15} - \frac{20470047056033004364923992987058072487023038665094339993923567654949459391704529134271}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{14} - \frac{21809361938587263457925817145540225220053165173977102080714825827831414402198722290863}{428531725790289027610889299374761654810522188603704570202869372689015726822902052052275} a^{13} + \frac{121575012821649160856178386555830168718114913477670993204102576083959654419740407461576}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{12} + \frac{49917892646606380943657989664715887477495810790976922121218585467909404533395749710986}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{11} + \frac{77519201904564882917648088890616669133960704547588491434258232394481932417255965803486}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{10} + \frac{547669303068319672907535106988595556051181841480996877837599147789724771982188803441528}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{9} + \frac{523075371475322972708544176796504599456912820456780207635053316606733452528522819907554}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{8} - \frac{3554054460336082640855383722050145120389163170857800753714429886696053094926925965519}{85706345158057805522177859874952330962104437720740914040573874537803145364580410410455} a^{7} - \frac{1002449576304064852799577302830820916053012513924215120869254088949964585446392921031291}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{6} - \frac{401041494394654429650159380819718354723722618335006081847472145599131759553082463150051}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{5} - \frac{4739803496302937471456168309537220439107704391520183929925146763921400802864651635031}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{4} + \frac{473995391951123900350641995154342335229161978848913899250393640657302482688185077966706}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{3} - \frac{157047141833508435187832833828538150246403437157494068082230922614962809354560852677498}{2142658628951445138054446496873808274052610943018522851014346863445078634114510260261375} a^{2} - \frac{91963375591408690998233413080738566183269333003753223256786578290721117539020890661476}{428531725790289027610889299374761654810522188603704570202869372689015726822902052052275} a - \frac{40618558021148624347477273679755739942470250024649283912433428190309514945929422056816}{85706345158057805522177859874952330962104437720740914040573874537803145364580410410455}$

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

Class group and class number

$C_{2856068}$, which has order $2856068$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 6698055135.71 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 6698055135.71 \cdot 2856068}{2\sqrt{685578251337344185848967768724165145970351361}}\approx 0.887355604634$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.119139761.1, 8.8.213496205355753436961.2

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
89Data not computed