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Properties

Label 16.0.557...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.573\times 10^{19}$
Root discriminant \(17.14\)
Ramified primes $2,3,5$
Class number $2$
Class group [2]
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 115*x^12 - 252*x^11 + 570*x^10 - 1164*x^9 + 1938*x^8 - 2496*x^7 + 2508*x^6 - 2016*x^5 + 1324*x^4 - 720*x^3 + 312*x^2 - 96*x + 16)
 
gp: K = bnfinit(y^16 - 6*y^15 + 21*y^14 - 54*y^13 + 115*y^12 - 252*y^11 + 570*y^10 - 1164*y^9 + 1938*y^8 - 2496*y^7 + 2508*y^6 - 2016*y^5 + 1324*y^4 - 720*y^3 + 312*y^2 - 96*y + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 115*x^12 - 252*x^11 + 570*x^10 - 1164*x^9 + 1938*x^8 - 2496*x^7 + 2508*x^6 - 2016*x^5 + 1324*x^4 - 720*x^3 + 312*x^2 - 96*x + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 115*x^12 - 252*x^11 + 570*x^10 - 1164*x^9 + 1938*x^8 - 2496*x^7 + 2508*x^6 - 2016*x^5 + 1324*x^4 - 720*x^3 + 312*x^2 - 96*x + 16)
 

\( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 115 x^{12} - 252 x^{11} + 570 x^{10} - 1164 x^{9} + \cdots + 16 \) 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:   \(55725627801600000000\) \(\medspace = 2^{28}\cdot 3^{12}\cdot 5^{8}\) 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:  \(17.14\)
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:  $2^{2}3^{3/4}5^{1/2}\approx 20.38853093816547$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{1880}a^{14}-\frac{11}{94}a^{13}-\frac{93}{1880}a^{12}-\frac{113}{470}a^{11}+\frac{41}{376}a^{10}+\frac{163}{940}a^{9}+\frac{19}{470}a^{8}+\frac{21}{235}a^{7}+\frac{61}{940}a^{6}-\frac{12}{235}a^{5}-\frac{77}{235}a^{4}-\frac{11}{235}a^{3}+\frac{27}{94}a^{2}+\frac{54}{235}a+\frac{63}{235}$, $\frac{1}{38872760}a^{15}-\frac{5127}{38872760}a^{14}+\frac{97907}{1253960}a^{13}+\frac{2961469}{38872760}a^{12}+\frac{6458979}{38872760}a^{11}+\frac{4204341}{38872760}a^{10}-\frac{360162}{4859095}a^{9}-\frac{3076647}{19436380}a^{8}-\frac{549837}{19436380}a^{7}-\frac{1885167}{3887276}a^{6}-\frac{3923211}{9718190}a^{5}-\frac{3104219}{9718190}a^{4}+\frac{77031}{335110}a^{3}+\frac{4012753}{9718190}a^{2}-\frac{81559}{971819}a-\frac{1340791}{4859095}$ 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

$C_{2}$, which has order $2$

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:   \( -\frac{303109}{413540} a^{15} + \frac{3242043}{827080} a^{14} - \frac{86059}{6670} a^{13} + \frac{26002727}{827080} a^{12} - \frac{6675397}{103385} a^{11} + \frac{119314017}{827080} a^{10} - \frac{6758405}{20677} a^{9} + \frac{267873567}{413540} a^{8} - \frac{104969489}{103385} a^{7} + \frac{495416007}{413540} a^{6} - \frac{22663621}{20677} a^{5} + \frac{82263402}{103385} a^{4} - \frac{1687087}{3565} a^{3} + \frac{47502621}{206770} a^{2} - \frac{8490162}{103385} a + \frac{1828634}{103385} \)  (order $6$) 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:   $\frac{9059081}{19436380}a^{15}-\frac{93006357}{38872760}a^{14}+\frac{51033}{6670}a^{13}-\frac{708349513}{38872760}a^{12}+\frac{179007323}{4859095}a^{11}-\frac{3238251863}{38872760}a^{10}+\frac{367546455}{1943638}a^{9}-\frac{152453109}{413540}a^{8}+\frac{2725219861}{4859095}a^{7}-\frac{12317660963}{19436380}a^{6}+\frac{541342928}{971819}a^{5}-\frac{3815787381}{9718190}a^{4}+\frac{38126073}{167555}a^{3}-\frac{1049922329}{9718190}a^{2}+\frac{185344238}{4859095}a-\frac{38122386}{4859095}$, $\frac{2309253}{19436380}a^{15}-\frac{19154211}{38872760}a^{14}+\frac{233604}{156745}a^{13}-\frac{129222089}{38872760}a^{12}+\frac{63669833}{9718190}a^{11}-\frac{613256979}{38872760}a^{10}+\frac{67193805}{1943638}a^{9}-\frac{623948987}{9718190}a^{8}+\frac{18625223}{206770}a^{7}-\frac{1792769909}{19436380}a^{6}+\frac{76070192}{971819}a^{5}-\frac{475501163}{9718190}a^{4}+\frac{4622499}{167555}a^{3}-\frac{128204707}{9718190}a^{2}+\frac{15726134}{4859095}a-\frac{6673948}{4859095}$, $\frac{359091}{1253960}a^{15}-\frac{434589}{313490}a^{14}+\frac{5465447}{1253960}a^{13}-\frac{3216781}{313490}a^{12}+\frac{25942657}{1253960}a^{11}-\frac{29826187}{626980}a^{10}+\frac{13424235}{125396}a^{9}-\frac{64538827}{313490}a^{8}+\frac{193924837}{626980}a^{7}-\frac{54365541}{156745}a^{6}+\frac{9722558}{31349}a^{5}-\frac{34577069}{156745}a^{4}+\frac{1350873}{10810}a^{3}-\frac{9084956}{156745}a^{2}+\frac{3026334}{156745}a-\frac{625943}{156745}$, $\frac{10486343}{38872760}a^{15}-\frac{10878821}{7774552}a^{14}+\frac{5640271}{1253960}a^{13}-\frac{419117581}{38872760}a^{12}+\frac{170292335}{7774552}a^{11}-\frac{1921796497}{38872760}a^{10}+\frac{2181322159}{19436380}a^{9}-\frac{4265388013}{19436380}a^{8}+\frac{6544111293}{19436380}a^{7}-\frac{7506973199}{19436380}a^{6}+\frac{3368996493}{9718190}a^{5}-\frac{2440209931}{9718190}a^{4}+\frac{9934449}{67022}a^{3}-\frac{682973351}{9718190}a^{2}+\frac{112988034}{4859095}a-\frac{4075516}{971819}$, $\frac{1803313}{3887276}a^{15}-\frac{12129147}{4859095}a^{14}+\frac{1046715}{125396}a^{13}-\frac{401158461}{19436380}a^{12}+\frac{834739811}{19436380}a^{11}-\frac{372721437}{3887276}a^{10}+\frac{1050331693}{4859095}a^{9}-\frac{8384473723}{19436380}a^{8}+\frac{6698722183}{9718190}a^{7}-\frac{8179163213}{9718190}a^{6}+\frac{7836275969}{9718190}a^{5}-\frac{2976925778}{4859095}a^{4}+\frac{63580884}{167555}a^{3}-\frac{184437484}{971819}a^{2}+\frac{338835571}{4859095}a-\frac{73774473}{4859095}$, $\frac{3382901}{19436380}a^{15}-\frac{28838467}{38872760}a^{14}+\frac{1426837}{626980}a^{13}-\frac{200248703}{38872760}a^{12}+\frac{199283537}{19436380}a^{11}-\frac{948265573}{38872760}a^{10}+\frac{208586167}{3887276}a^{9}-\frac{1958091633}{19436380}a^{8}+\frac{1405723997}{9718190}a^{7}-\frac{2984654303}{19436380}a^{6}+\frac{260419263}{1943638}a^{5}-\frac{848437761}{9718190}a^{4}+\frac{8239603}{167555}a^{3}-\frac{228673529}{9718190}a^{2}+\frac{28594393}{4859095}a-\frac{6337416}{4859095}$, $\frac{6927013}{9718190}a^{15}-\frac{142200241}{38872760}a^{14}+\frac{7306437}{626980}a^{13}-\frac{1075711841}{38872760}a^{12}+\frac{1084230001}{19436380}a^{11}-\frac{4904327763}{38872760}a^{10}+\frac{2787325009}{9718190}a^{9}-\frac{2168573899}{3887276}a^{8}+\frac{4105356422}{4859095}a^{7}-\frac{18385402617}{19436380}a^{6}+\frac{3986551783}{4859095}a^{5}-\frac{553706402}{971819}a^{4}+\frac{53614399}{167555}a^{3}-\frac{1415905669}{9718190}a^{2}+\frac{217279722}{4859095}a-\frac{28378308}{4859095}$ Copy content Toggle raw display
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:  \( 5178.51305794 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5178.51305794 \cdot 2}{6\cdot\sqrt{55725627801600000000}}\cr\approx \mathstrut & 0.561688177496 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 21*x^14 - 54*x^13 + 115*x^12 - 252*x^11 + 570*x^10 - 1164*x^9 + 1938*x^8 - 2496*x^7 + 2508*x^6 - 2016*x^5 + 1324*x^4 - 720*x^3 + 312*x^2 - 96*x + 16)
 
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 - 6*x^15 + 21*x^14 - 54*x^13 + 115*x^12 - 252*x^11 + 570*x^10 - 1164*x^9 + 1938*x^8 - 2496*x^7 + 2508*x^6 - 2016*x^5 + 1324*x^4 - 720*x^3 + 312*x^2 - 96*x + 16, 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 - 6*x^15 + 21*x^14 - 54*x^13 + 115*x^12 - 252*x^11 + 570*x^10 - 1164*x^9 + 1938*x^8 - 2496*x^7 + 2508*x^6 - 2016*x^5 + 1324*x^4 - 720*x^3 + 312*x^2 - 96*x + 16);
 
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 - 6*x^15 + 21*x^14 - 54*x^13 + 115*x^12 - 252*x^11 + 570*x^10 - 1164*x^9 + 1938*x^8 - 2496*x^7 + 2508*x^6 - 2016*x^5 + 1324*x^4 - 720*x^3 + 312*x^2 - 96*x + 16);
 
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

$C_2^2\times D_4$ (as 16T25):

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 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.0.21600.2, 4.0.5400.2, 4.0.21600.1, 4.0.5400.1, \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), 8.0.207360000.1, 8.0.7464960000.4, 8.0.7464960000.1, 8.0.466560000.2, 8.0.29160000.1, 8.0.7464960000.3, 8.0.7464960000.2

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 16 siblings: 16.0.1426576071720960000.1, 16.0.3482851737600000000.5, 16.0.891610044825600000000.13, 16.0.891610044825600000000.7, 16.0.891610044825600000000.14, 16.0.891610044825600000000.8, 16.8.891610044825600000000.2
Minimal sibling: 16.0.1426576071720960000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(2\) Copy content Toggle raw display 2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) Copy content Toggle raw display 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$