↑

Properties

Label 16.0.818...809.2
Degree $16$
Signature $[0, 8]$
Discriminant $8.186\times 10^{26}$
Root discriminant \(48.09\)
Ramified primes $17,67$
Class number $20$ (GRH)
Class group [20] (GRH)
Galois group $C_4\wr C_2$ (as 16T42)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 - 126*x^10 + 938*x^9 - 2071*x^8 + 2744*x^7 - 1370*x^6 - 1710*x^5 + 8085*x^4 - 11910*x^3 + 8237*x^2 - 2756*x + 1283)
 
gp: K = bnfinit(y^16 - 8*y^15 + 32*y^14 - 84*y^13 + 146*y^12 - 148*y^11 - 126*y^10 + 938*y^9 - 2071*y^8 + 2744*y^7 - 1370*y^6 - 1710*y^5 + 8085*y^4 - 11910*y^3 + 8237*y^2 - 2756*y + 1283, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 - 126*x^10 + 938*x^9 - 2071*x^8 + 2744*x^7 - 1370*x^6 - 1710*x^5 + 8085*x^4 - 11910*x^3 + 8237*x^2 - 2756*x + 1283);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 - 126*x^10 + 938*x^9 - 2071*x^8 + 2744*x^7 - 1370*x^6 - 1710*x^5 + 8085*x^4 - 11910*x^3 + 8237*x^2 - 2756*x + 1283)
 

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 146 x^{12} - 148 x^{11} - 126 x^{10} + 938 x^{9} + \cdots + 1283 \) 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:   \(818629961123679547712831809\) \(\medspace = 17^{10}\cdot 67^{8}\) 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:  \(48.09\)
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:  $17^{3/4}67^{1/2}\approx 68.52895233095656$
Ramified primes:   \(17\), \(67\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{15837882754}a^{14}-\frac{7}{15837882754}a^{13}+\frac{1100670395}{7918941377}a^{12}+\frac{2629838105}{15837882754}a^{11}+\frac{842522104}{7918941377}a^{10}+\frac{1783342131}{15837882754}a^{9}-\frac{8242490}{465820081}a^{8}+\frac{3511262541}{15837882754}a^{7}-\frac{4695636869}{15837882754}a^{6}+\frac{1266172532}{7918941377}a^{5}+\frac{67724917}{688603598}a^{4}-\frac{130482117}{7918941377}a^{3}-\frac{1087613311}{7918941377}a^{2}+\frac{7349109215}{15837882754}a-\frac{2012044641}{15837882754}$, $\frac{1}{123044511115826}a^{15}+\frac{3877}{123044511115826}a^{14}+\frac{12006215784333}{61522255557913}a^{13}-\frac{24485185958379}{123044511115826}a^{12}-\frac{9431188246158}{61522255557913}a^{11}+\frac{3467261586738}{61522255557913}a^{10}-\frac{9318061086406}{61522255557913}a^{9}-\frac{3413127761737}{123044511115826}a^{8}+\frac{29660985419423}{123044511115826}a^{7}-\frac{31198628288281}{123044511115826}a^{6}-\frac{15756832628797}{123044511115826}a^{5}-\frac{39888037606767}{123044511115826}a^{4}-\frac{2496602348977}{123044511115826}a^{3}+\frac{13117906599625}{61522255557913}a^{2}+\frac{33530861213929}{123044511115826}a+\frac{20063244855357}{61522255557913}$ 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

$C_{20}$, which has order $20$ (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:   $\frac{36215956017}{61522255557913}a^{15}-\frac{249450094457}{61522255557913}a^{14}+\frac{1706950979893}{123044511115826}a^{13}-\frac{3814357478823}{123044511115826}a^{12}+\frac{111596637455}{2674880676431}a^{11}-\frac{1147589142195}{61522255557913}a^{10}-\frac{15698615580027}{123044511115826}a^{9}+\frac{53823942994027}{123044511115826}a^{8}-\frac{83023339856125}{123044511115826}a^{7}+\frac{36835696785648}{61522255557913}a^{6}+\frac{18183871644284}{61522255557913}a^{5}-\frac{153597956484833}{123044511115826}a^{4}+\frac{215118622083967}{61522255557913}a^{3}-\frac{157387161713235}{61522255557913}a^{2}-\frac{9748517609480}{61522255557913}a+\frac{10578657642064}{61522255557913}$, $\frac{8661168467}{61522255557913}a^{15}-\frac{26812184949}{61522255557913}a^{14}+\frac{12542542775}{123044511115826}a^{13}+\frac{13667844579}{5349761352862}a^{12}-\frac{687539136549}{61522255557913}a^{11}+\frac{1424270776683}{61522255557913}a^{10}-\frac{6115476303677}{123044511115826}a^{9}+\frac{3095337090829}{123044511115826}a^{8}+\frac{13318885223249}{123044511115826}a^{7}-\frac{15802443247149}{61522255557913}a^{6}+\frac{31343493591902}{61522255557913}a^{5}-\frac{20771550787517}{123044511115826}a^{4}+\frac{34244866560431}{61522255557913}a^{3}+\frac{130173207571122}{61522255557913}a^{2}-\frac{46395509244007}{61522255557913}a+\frac{21588152259767}{61522255557913}$, $\frac{3550146878}{61522255557913}a^{15}-\frac{26626101585}{61522255557913}a^{14}+\frac{110026870553}{61522255557913}a^{13}-\frac{311345451222}{61522255557913}a^{12}+\frac{542315537088}{61522255557913}a^{11}-\frac{892719493523}{123044511115826}a^{10}-\frac{2663655829909}{123044511115826}a^{9}+\frac{6107490036171}{61522255557913}a^{8}-\frac{12883856637981}{61522255557913}a^{7}+\frac{35303373990061}{123044511115826}a^{6}-\frac{18262966585019}{123044511115826}a^{5}-\frac{15871908692793}{123044511115826}a^{4}+\frac{2169162494142}{2674880676431}a^{3}-\frac{61872068465308}{61522255557913}a^{2}+\frac{92262495756555}{123044511115826}a-\frac{13248766436273}{61522255557913}$, $\frac{16473265596}{61522255557913}a^{15}-\frac{123549491970}{61522255557913}a^{14}+\frac{442463198423}{61522255557913}a^{13}-\frac{2004353656409}{123044511115826}a^{12}+\frac{1333743192544}{61522255557913}a^{11}-\frac{868154328283}{123044511115826}a^{10}-\frac{3976052369861}{61522255557913}a^{9}+\frac{26891370811833}{123044511115826}a^{8}-\frac{19861949457755}{61522255557913}a^{7}+\frac{13412337133177}{61522255557913}a^{6}+\frac{21667608638545}{123044511115826}a^{5}-\frac{34238449720927}{61522255557913}a^{4}+\frac{6108578283969}{5349761352862}a^{3}-\frac{143756271236919}{123044511115826}a^{2}+\frac{34073223226026}{61522255557913}a-\frac{6145994677640}{61522255557913}$, $\frac{15191375697}{61522255557913}a^{15}-\frac{225957690123}{123044511115826}a^{14}+\frac{399733340552}{61522255557913}a^{13}-\frac{913231513251}{61522255557913}a^{12}+\frac{2843080501957}{123044511115826}a^{11}-\frac{1821069652265}{123044511115826}a^{10}-\frac{6818587241747}{123044511115826}a^{9}+\frac{14218046978722}{61522255557913}a^{8}-\frac{25461156770963}{61522255557913}a^{7}+\frac{51622590557649}{123044511115826}a^{6}-\frac{14995964947407}{61522255557913}a^{5}-\frac{46510629743704}{61522255557913}a^{4}+\frac{129814143673207}{61522255557913}a^{3}-\frac{121303960345375}{61522255557913}a^{2}+\frac{51803559335599}{61522255557913}a-\frac{16678449812475}{123044511115826}$, $\frac{487592516}{61522255557913}a^{15}+\frac{23682866340}{61522255557913}a^{14}-\frac{116825167187}{61522255557913}a^{13}+\frac{194085607636}{61522255557913}a^{12}-\frac{272346534602}{61522255557913}a^{11}+\frac{260266170705}{61522255557913}a^{10}-\frac{193291308503}{61522255557913}a^{9}-\frac{420656771112}{61522255557913}a^{8}-\frac{1430695708730}{61522255557913}a^{7}+\frac{21501732227667}{61522255557913}a^{6}-\frac{19244115163925}{61522255557913}a^{5}+\frac{11810831463612}{61522255557913}a^{4}-\frac{43197085827923}{61522255557913}a^{3}-\frac{84131122762883}{61522255557913}a^{2}+\frac{124199517949667}{61522255557913}a-\frac{28749223150559}{61522255557913}$, $\frac{107836843815}{61522255557913}a^{15}-\frac{681762248920}{61522255557913}a^{14}+\frac{2188923777022}{61522255557913}a^{13}-\frac{4682812760198}{61522255557913}a^{12}+\frac{5880068692570}{61522255557913}a^{11}-\frac{2339181354869}{61522255557913}a^{10}-\frac{20696580670176}{61522255557913}a^{9}+\frac{63126965492558}{61522255557913}a^{8}-\frac{90943453451471}{61522255557913}a^{7}+\frac{83872283149105}{61522255557913}a^{6}+\frac{52876177084179}{61522255557913}a^{5}-\frac{117981270910366}{61522255557913}a^{4}+\frac{549255149499390}{61522255557913}a^{3}-\frac{212141881848969}{61522255557913}a^{2}+\frac{141524077184018}{61522255557913}a-\frac{33624390391298}{61522255557913}$ 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:  \( 496177.351463 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 496177.351463 \cdot 20}{2\cdot\sqrt{818629961123679547712831809}}\cr\approx \mathstrut & 0.421242387647 \end{aligned}\] (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 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 - 126*x^10 + 938*x^9 - 2071*x^8 + 2744*x^7 - 1370*x^6 - 1710*x^5 + 8085*x^4 - 11910*x^3 + 8237*x^2 - 2756*x + 1283)
 
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 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 - 126*x^10 + 938*x^9 - 2071*x^8 + 2744*x^7 - 1370*x^6 - 1710*x^5 + 8085*x^4 - 11910*x^3 + 8237*x^2 - 2756*x + 1283, 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 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 - 126*x^10 + 938*x^9 - 2071*x^8 + 2744*x^7 - 1370*x^6 - 1710*x^5 + 8085*x^4 - 11910*x^3 + 8237*x^2 - 2756*x + 1283);
 
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 + 32*x^14 - 84*x^13 + 146*x^12 - 148*x^11 - 126*x^10 + 938*x^9 - 2071*x^8 + 2744*x^7 - 1370*x^6 - 1710*x^5 + 8085*x^4 - 11910*x^3 + 8237*x^2 - 2756*x + 1283);
 
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

$C_4\wr C_2$ (as 16T42):

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 32
The 14 conjugacy class representatives for $C_4\wr C_2$
Character table for $C_4\wr C_2$

Intermediate fields

\(\Q(\sqrt{-67}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1139}) \), 4.0.76313.1 x2, 4.2.19363.1 x2, \(\Q(\sqrt{17}, \sqrt{-67})\), 8.0.28611710209697.1, 8.0.99002457473.1, 8.0.1683041777041.2

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

Galois closure: deg 32
Degree 8 siblings: 8.0.99002457473.1, 8.0.28611710209697.1
Degree 16 sibling: 16.4.52703064995487500398531609.8
Minimal sibling: 8.0.99002457473.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

# 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 17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.3.1$x^{4} + 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} + 17$$4$$1$$3$$C_4$$[\ ]_{4}$
\(67\) Copy content Toggle raw display 67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$