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Properties

Label 16.0.400...969.19
Degree $16$
Signature $[0, 8]$
Discriminant $4.009\times 10^{30}$
Root discriminant \(81.79\)
Ramified primes $17,47$
Class number $5760$ (GRH)
Class group [4, 12, 120] (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824)
 
gp: K = bnfinit(y^16 - 8*y^15 + 64*y^14 - 308*y^13 + 1682*y^12 - 6452*y^11 + 25776*y^10 - 76124*y^9 + 281589*y^8 - 729716*y^7 + 1987704*y^6 - 3669248*y^5 + 6950160*y^4 - 8513728*y^3 + 13870848*y^2 - 10122240*y + 82509824, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824)
 

\( x^{16} - 8 x^{15} + 64 x^{14} - 308 x^{13} + 1682 x^{12} - 6452 x^{11} + 25776 x^{10} - 76124 x^{9} + \cdots + 82509824 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4009292695690170390860412175969\) \(\medspace = 17^{14}\cdot 47^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(81.79\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $17^{7/8}47^{1/2}\approx 81.7884043055344$
Ramified primes:   \(17\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{16}a^{7}+\frac{3}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{2176}a^{8}-\frac{1}{544}a^{7}+\frac{29}{1088}a^{6}+\frac{7}{136}a^{5}-\frac{87}{2176}a^{4}+\frac{41}{544}a^{3}-\frac{57}{544}a^{2}-\frac{1}{136}a+\frac{2}{17}$, $\frac{1}{2176}a^{9}+\frac{21}{1088}a^{7}-\frac{1}{34}a^{6}-\frac{47}{2176}a^{5}-\frac{3}{136}a^{4}-\frac{63}{544}a^{3}+\frac{27}{136}a^{2}+\frac{3}{34}a+\frac{8}{17}$, $\frac{1}{4352}a^{10}-\frac{1}{4352}a^{9}-\frac{37}{2176}a^{7}+\frac{29}{4352}a^{6}-\frac{81}{4352}a^{5}+\frac{93}{2176}a^{4}+\frac{47}{1088}a^{3}-\frac{91}{544}a^{2}+\frac{47}{136}a+\frac{5}{17}$, $\frac{1}{4352}a^{11}-\frac{1}{4352}a^{9}-\frac{69}{4352}a^{7}-\frac{7}{272}a^{6}+\frac{233}{4352}a^{5}-\frac{5}{272}a^{4}-\frac{25}{1088}a^{3}-\frac{5}{68}a^{2}-\frac{9}{68}a-\frac{6}{17}$, $\frac{1}{193089536}a^{12}-\frac{3}{96544768}a^{11}+\frac{1509}{193089536}a^{10}-\frac{3745}{96544768}a^{9}+\frac{36375}{193089536}a^{8}-\frac{50313}{96544768}a^{7}-\frac{115953}{11358208}a^{6}+\frac{3117201}{96544768}a^{5}+\frac{2112529}{48272384}a^{4}-\frac{3424169}{24136192}a^{3}-\frac{556097}{3017024}a^{2}+\frac{393285}{1508512}a+\frac{10123}{47141}$, $\frac{1}{193089536}a^{13}+\frac{1473}{193089536}a^{11}+\frac{23}{2839552}a^{10}-\frac{8565}{193089536}a^{9}+\frac{3611}{24136192}a^{8}-\frac{130589}{11358208}a^{7}+\frac{332151}{48272384}a^{6}-\frac{17887}{12068096}a^{5}-\frac{323557}{12068096}a^{4}+\frac{924425}{12068096}a^{3}-\frac{174125}{1508512}a^{2}+\frac{12705}{44368}a+\frac{8051}{47141}$, $\frac{1}{386179072}a^{14}-\frac{1}{386179072}a^{13}-\frac{1}{386179072}a^{12}+\frac{8935}{386179072}a^{11}-\frac{15995}{386179072}a^{10}+\frac{74449}{386179072}a^{9}-\frac{50707}{386179072}a^{8}+\frac{10958573}{386179072}a^{7}-\frac{1217365}{193089536}a^{6}-\frac{4685481}{96544768}a^{5}+\frac{42137}{1027072}a^{4}+\frac{123929}{754256}a^{3}+\frac{36773}{177472}a^{2}+\frac{97335}{377128}a+\frac{585}{2773}$, $\frac{1}{52078951112704}a^{15}+\frac{67421}{52078951112704}a^{14}+\frac{101203}{52078951112704}a^{13}-\frac{17135}{52078951112704}a^{12}-\frac{1185332487}{52078951112704}a^{11}+\frac{4475051079}{52078951112704}a^{10}-\frac{11739904735}{52078951112704}a^{9}-\frac{7485817637}{52078951112704}a^{8}+\frac{569849240177}{26039475556352}a^{7}+\frac{1035857207}{95733366016}a^{6}+\frac{30792575357}{3254934444544}a^{5}-\frac{2435070743}{191466732032}a^{4}+\frac{2060219389}{27584190208}a^{3}+\frac{8842388027}{101716701392}a^{2}-\frac{3696900581}{101716701392}a+\frac{1396154377}{6357293837}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{4}\times C_{12}\times C_{120}$, which has order $5760$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1}{48272384}a^{14}-\frac{7}{48272384}a^{13}+\frac{61}{96544768}a^{12}-\frac{23}{12068096}a^{11}-\frac{1}{1636352}a^{10}+\frac{103}{6034048}a^{9}-\frac{19901}{96544768}a^{8}+\frac{8533}{12068096}a^{7}-\frac{225587}{96544768}a^{6}+\frac{13123}{2839552}a^{5}-\frac{4475}{513536}a^{4}+\frac{126825}{12068096}a^{3}-\frac{25579}{1508512}a^{2}+\frac{9323}{754256}a-\frac{53}{47141}$, $\frac{26211975}{1531733856256}a^{15}-\frac{739336805}{6509868889088}a^{14}+\frac{7384793237}{13019737778176}a^{13}-\frac{3521896769}{6509868889088}a^{12}+\frac{2479312567}{3254934444544}a^{11}+\frac{28684164935}{6509868889088}a^{10}+\frac{246960055367}{13019737778176}a^{9}+\frac{27156417167}{138507848704}a^{8}-\frac{16883117400623}{26039475556352}a^{7}+\frac{11301183311499}{3254934444544}a^{6}-\frac{22176491901985}{3254934444544}a^{5}+\frac{14510409323801}{813733611136}a^{4}-\frac{29093025962531}{1627467222272}a^{3}+\frac{2717149439689}{101716701392}a^{2}-\frac{20927552505}{5983335376}a+\frac{1822297301381}{6357293837}$, $\frac{187283161}{26039475556352}a^{15}-\frac{27921357}{277015697408}a^{14}+\frac{2991940131}{3254934444544}a^{13}-\frac{77074421099}{13019737778176}a^{12}+\frac{412639206473}{13019737778176}a^{11}-\frac{1824796064273}{13019737778176}a^{10}+\frac{1707188880407}{3254934444544}a^{9}-\frac{20713101891905}{13019737778176}a^{8}+\frac{104875222548557}{26039475556352}a^{7}-\frac{6778013271997}{813733611136}a^{6}+\frac{45747417504597}{3254934444544}a^{5}-\frac{15880393416421}{813733611136}a^{4}+\frac{44187369257933}{1627467222272}a^{3}-\frac{4970134094517}{101716701392}a^{2}+\frac{9929451063767}{101716701392}a-\frac{584057177107}{6357293837}$, $\frac{847033863}{26039475556352}a^{15}-\frac{5755213683}{13019737778176}a^{14}+\frac{51802024707}{13019737778176}a^{13}-\frac{81460483173}{3254934444544}a^{12}+\frac{428934262665}{3254934444544}a^{11}-\frac{3706355938205}{6509868889088}a^{10}+\frac{27186961497739}{13019737778176}a^{9}-\frac{9960613358917}{1627467222272}a^{8}+\frac{390750542529569}{26039475556352}a^{7}-\frac{387513214222867}{13019737778176}a^{6}+\frac{328242294940373}{6509868889088}a^{5}-\frac{235515757914257}{3254934444544}a^{4}+\frac{1481532239979}{13792095104}a^{3}-\frac{36536899112773}{203433402784}a^{2}+\frac{17314068096715}{50858350696}a-\frac{1496649527115}{6357293837}$, $\frac{91676771}{26039475556352}a^{15}-\frac{677933507}{26039475556352}a^{14}+\frac{3197006203}{26039475556352}a^{13}-\frac{18122869101}{26039475556352}a^{12}+\frac{82920931609}{26039475556352}a^{11}-\frac{290396732303}{26039475556352}a^{10}+\frac{853382261141}{26039475556352}a^{9}-\frac{3601837446327}{26039475556352}a^{8}+\frac{288823749639}{813733611136}a^{7}-\frac{13818804730897}{13019737778176}a^{6}+\frac{11437268737801}{6509868889088}a^{5}-\frac{14815200804541}{3254934444544}a^{4}+\frac{12578695971287}{1627467222272}a^{3}-\frac{3377487470669}{203433402784}a^{2}+\frac{3654051376713}{101716701392}a-\frac{708141239337}{6357293837}$, $\frac{496250441}{26039475556352}a^{15}-\frac{4256168045}{13019737778176}a^{14}+\frac{10288937265}{3254934444544}a^{13}-\frac{141862351707}{6509868889088}a^{12}+\frac{1553315973823}{13019737778176}a^{11}-\frac{1784183906197}{3254934444544}a^{10}+\frac{13701260073347}{6509868889088}a^{9}-\frac{43467527578635}{6509868889088}a^{8}+\frac{446231656056897}{26039475556352}a^{7}-\frac{478943229532383}{13019737778176}a^{6}+\frac{420064421408353}{6509868889088}a^{5}-\frac{322036556385973}{3254934444544}a^{4}+\frac{112366927824423}{813733611136}a^{3}-\frac{47620317683867}{203433402784}a^{2}+\frac{22627967986451}{50858350696}a-\frac{3990023841551}{6357293837}$, $\frac{91676771}{26039475556352}a^{15}-\frac{348609029}{13019737778176}a^{14}+\frac{832999515}{6509868889088}a^{13}-\frac{1729596603}{13019737778176}a^{12}-\frac{3408008951}{13019737778176}a^{11}+\frac{31669084203}{13019737778176}a^{10}-\frac{22371558157}{6509868889088}a^{9}+\frac{804731928039}{13019737778176}a^{8}-\frac{6938256084765}{26039475556352}a^{7}+\frac{6904682992971}{6509868889088}a^{6}-\frac{2073591974129}{813733611136}a^{5}+\frac{10394879382579}{1627467222272}a^{4}-\frac{12441769615383}{1627467222272}a^{3}+\frac{1493320931545}{101716701392}a^{2}+\frac{1190615051431}{101716701392}a+\frac{32891599389}{373958461}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 2668614326.25 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2668614326.25 \cdot 5760}{2\cdot\sqrt{4009292695690170390860412175969}}\cr\approx \mathstrut & 9323.59427240 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 64*x^14 - 308*x^13 + 1682*x^12 - 6452*x^11 + 25776*x^10 - 76124*x^9 + 281589*x^8 - 729716*x^7 + 1987704*x^6 - 3669248*x^5 + 6950160*x^4 - 8513728*x^3 + 13870848*x^2 - 10122240*x + 82509824);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$\SD_{16}$ (as 16T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-799}) \), \(\Q(\sqrt{17}, \sqrt{-47})\), 4.2.230911.1 x2, 4.0.10852817.1 x2, 8.0.117783636835489.1, 8.2.42602592046879.1 x4

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 8 sibling: 8.2.42602592046879.1
Minimal sibling: 8.2.42602592046879.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{16}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ R ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(17\) Copy content Toggle raw display 17.8.7.3$x^{8} + 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} + 17$$8$$1$$7$$C_8$$[\ ]_{8}$
\(47\) Copy content Toggle raw display 47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$