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Properties

Label 16.0.934...456.9
Degree $16$
Signature $[0, 8]$
Discriminant $9.349\times 10^{18}$
Root discriminant \(15.33\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $D_4\times C_2$ (as 16T9)

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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 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25)
 
gp: K = bnfinit(y^16 - 8*y^15 + 48*y^14 - 196*y^13 + 642*y^12 - 1668*y^11 + 3580*y^10 - 6328*y^9 + 9297*y^8 - 11276*y^7 + 11224*y^6 - 9024*y^5 + 5736*y^4 - 2780*y^3 + 972*y^2 - 220*y + 25, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25)
 

\( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) 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:   \(9349208943630483456\) \(\medspace = 2^{44}\cdot 3^{12}\) 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:  \(15.33\)
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:  $2^{11/4}3^{3/4}\approx 15.33463450191054$
Ramified primes:   \(2\), \(3\) 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 a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{5}a^{8}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{65}a^{14}+\frac{6}{65}a^{13}-\frac{2}{65}a^{12}-\frac{14}{65}a^{11}+\frac{12}{65}a^{10}+\frac{12}{65}a^{9}-\frac{31}{65}a^{8}-\frac{11}{65}a^{7}+\frac{24}{65}a^{6}-\frac{2}{65}a^{5}-\frac{3}{13}a^{4}-\frac{9}{65}a^{3}-\frac{24}{65}a^{2}-\frac{12}{65}a-\frac{2}{13}$, $\frac{1}{17095}a^{15}+\frac{124}{17095}a^{14}-\frac{241}{3419}a^{13}+\frac{340}{3419}a^{12}+\frac{3859}{17095}a^{11}-\frac{4604}{17095}a^{10}+\frac{8379}{17095}a^{9}-\frac{1506}{3419}a^{8}-\frac{12}{263}a^{7}+\frac{4377}{17095}a^{6}-\frac{1657}{3419}a^{5}-\frac{674}{17095}a^{4}-\frac{6962}{17095}a^{3}-\frac{738}{17095}a^{2}+\frac{8493}{17095}a-\frac{1588}{3419}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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:   \( \frac{1229}{1315} a^{15} - \frac{141262}{17095} a^{14} + \frac{856169}{17095} a^{13} - \frac{3642449}{17095} a^{12} + \frac{12106306}{17095} a^{11} - \frac{32236863}{17095} a^{10} + \frac{70027507}{17095} a^{9} - \frac{25115203}{3419} a^{8} + \frac{185376534}{17095} a^{7} - \frac{225133368}{17095} a^{6} + \frac{221440773}{17095} a^{5} - \frac{173497381}{17095} a^{4} + \frac{104335813}{17095} a^{3} - \frac{45658062}{17095} a^{2} + \frac{12999216}{17095} a - \frac{376745}{3419} \)  (order $24$) 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:   $\frac{2458}{1315}a^{15}-\frac{3687}{263}a^{14}+\frac{21727}{263}a^{13}-\frac{85306}{263}a^{12}+\frac{1359511}{1315}a^{11}-\frac{680119}{263}a^{10}+\frac{1407902}{263}a^{9}-\frac{2378595}{263}a^{8}+\frac{16627679}{1315}a^{7}-\frac{3792435}{263}a^{6}+\frac{17527984}{1315}a^{5}-\frac{2560316}{263}a^{4}+\frac{7165197}{1315}a^{3}-\frac{578303}{263}a^{2}+\frac{758661}{1315}a-\frac{19583}{263}$, $\frac{10653}{17095}a^{15}-\frac{90286}{17095}a^{14}+\frac{41769}{1315}a^{13}-\frac{2265299}{17095}a^{12}+\frac{7445664}{17095}a^{11}-\frac{19517298}{17095}a^{10}+\frac{41828911}{17095}a^{9}-\frac{73786477}{17095}a^{8}+\frac{107083488}{17095}a^{7}-\frac{127455201}{17095}a^{6}+\frac{122505991}{17095}a^{5}-\frac{93402042}{17095}a^{4}+\frac{54353761}{17095}a^{3}-\frac{22866327}{17095}a^{2}+\frac{1238306}{3419}a-\frac{168861}{3419}$, $\frac{5322}{3419}a^{15}-\frac{39915}{3419}a^{14}+\frac{1161581}{17095}a^{13}-\frac{347953}{1315}a^{12}+\frac{2836383}{3419}a^{11}-\frac{34902406}{17095}a^{10}+\frac{70486206}{17095}a^{9}-\frac{115786188}{17095}a^{8}+\frac{155569131}{17095}a^{7}-\frac{168936156}{17095}a^{6}+\frac{145782586}{17095}a^{5}-\frac{97273298}{17095}a^{4}+\frac{3677283}{1315}a^{3}-\frac{1234196}{1315}a^{2}+\frac{3198017}{17095}a-\frac{52807}{3419}$, $\frac{24761}{17095}a^{15}-\frac{14093}{1315}a^{14}+\frac{1071807}{17095}a^{13}-\frac{4161162}{17095}a^{12}+\frac{13119294}{17095}a^{11}-\frac{32365496}{17095}a^{10}+\frac{65945018}{17095}a^{9}-\frac{21851177}{3419}a^{8}+\frac{149132923}{17095}a^{7}-\frac{165041163}{17095}a^{6}+\frac{29364288}{3419}a^{5}-\frac{20408692}{3419}a^{4}+\frac{53485156}{17095}a^{3}-\frac{3943445}{3419}a^{2}+\frac{4547906}{17095}a-\frac{93492}{3419}$, $\frac{65128}{17095}a^{15}-\frac{486093}{17095}a^{14}+\frac{571491}{3419}a^{13}-\frac{11178429}{17095}a^{12}+\frac{2731182}{1315}a^{11}-\frac{1360691}{263}a^{10}+\frac{182213897}{17095}a^{9}-\frac{306163404}{17095}a^{8}+\frac{85029463}{3419}a^{7}-\frac{37001666}{1315}a^{6}+\frac{33856459}{1315}a^{5}-\frac{63516445}{3419}a^{4}+\frac{174957137}{17095}a^{3}-\frac{69284356}{17095}a^{2}+\frac{3550021}{3419}a-\frac{448244}{3419}$, $\frac{3505}{3419}a^{15}-\frac{157606}{17095}a^{14}+\frac{954913}{17095}a^{13}-\frac{4085056}{17095}a^{12}+\frac{13580371}{17095}a^{11}-\frac{7249198}{3419}a^{10}+\frac{78703672}{17095}a^{9}-\frac{141206648}{17095}a^{8}+\frac{41627362}{3419}a^{7}-\frac{50508292}{3419}a^{6}+\frac{247711332}{17095}a^{5}-\frac{193783391}{17095}a^{4}+\frac{116210564}{17095}a^{3}-\frac{51025844}{17095}a^{2}+\frac{14641708}{17095}a-\frac{34278}{263}$, $\frac{16873}{17095}a^{15}-\frac{139829}{17095}a^{14}+\frac{834801}{17095}a^{13}-\frac{3438457}{17095}a^{12}+\frac{11191332}{17095}a^{11}-\frac{28980163}{17095}a^{10}+\frac{4719479}{1315}a^{9}-\frac{106675661}{17095}a^{8}+\frac{152316293}{17095}a^{7}-\frac{177735046}{17095}a^{6}+\frac{166774197}{17095}a^{5}-\frac{24637686}{3419}a^{4}+\frac{68816493}{17095}a^{3}-\frac{27282844}{17095}a^{2}+\frac{6836949}{17095}a-\frac{164785}{3419}$ 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:  \( 10073.1288883 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10073.1288883 \cdot 1}{24\cdot\sqrt{9349208943630483456}}\cr\approx \mathstrut & 0.333429992862 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25)
 
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 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25, 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 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25);
 
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 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25);
 
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

$C_2\times D_4$ (as 16T9):

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 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.13824.1 x2, 4.0.13824.1 x2, 4.0.27648.1 x2, 4.4.27648.1 x2, \(\Q(\zeta_{24})\), 8.0.191102976.5 x2, 8.0.3057647616.6 x2, 8.0.764411904.5 x2, 8.0.3057647616.5 x2, 8.0.3057647616.7, 8.8.3057647616.1

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 siblings: 8.0.191102976.5, 8.0.3057647616.5, 8.0.764411904.5, 8.0.3057647616.6
Minimal sibling: 8.0.191102976.5

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 ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(2\) Copy content Toggle raw display 2.16.44.3$x^{16} + 16 x^{15} + 112 x^{14} + 476 x^{13} + 1462 x^{12} + 3532 x^{11} + 6880 x^{10} + 10824 x^{9} + 13685 x^{8} + 13460 x^{7} + 9452 x^{6} + 1528 x^{5} - 3900 x^{4} - 1068 x^{3} + 992 x^{2} - 92 x + 73$$8$$2$$44$$D_4\times C_2$$[2, 3, 7/2]^{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}$