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Properties

Label 16.0.235...056.3
Degree $16$
Signature $[0, 8]$
Discriminant $2.360\times 10^{19}$
Root discriminant \(16.25\)
Ramified primes $2,3,11$
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 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9)
 
gp: K = bnfinit(y^16 - 2*y^15 - y^14 + 18*y^13 - 27*y^12 - 4*y^11 + 62*y^10 - 88*y^9 + 62*y^8 - 112*y^7 + 294*y^6 - 444*y^5 + 277*y^4 + 42*y^3 - 33*y^2 - 18*y + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9)
 

\( x^{16} - 2 x^{15} - x^{14} + 18 x^{13} - 27 x^{12} - 4 x^{11} + 62 x^{10} - 88 x^{9} + 62 x^{8} + \cdots + 9 \) 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:   \(23595621172490797056\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 11^{8}\) 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:  \(16.25\)
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:  $2^{3/2}3^{1/2}11^{1/2}\approx 16.24807680927192$
Ramified primes:   \(2\), \(3\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{12}a^{12}+\frac{1}{12}a^{10}+\frac{1}{6}a^{9}-\frac{1}{4}a^{8}-\frac{1}{12}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{3}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{2412}a^{13}-\frac{2}{67}a^{12}+\frac{247}{2412}a^{11}+\frac{113}{2412}a^{10}+\frac{71}{804}a^{9}+\frac{49}{804}a^{8}-\frac{409}{2412}a^{7}+\frac{125}{804}a^{6}-\frac{89}{268}a^{5}-\frac{1123}{2412}a^{4}+\frac{853}{2412}a^{3}+\frac{121}{804}a^{2}+\frac{169}{402}a+\frac{15}{268}$, $\frac{1}{9648}a^{14}+\frac{11}{1206}a^{12}+\frac{205}{4824}a^{11}-\frac{55}{1072}a^{10}-\frac{11}{268}a^{9}-\frac{1885}{9648}a^{8}-\frac{323}{1608}a^{7}-\frac{781}{3216}a^{6}+\frac{527}{2412}a^{5}+\frac{3211}{9648}a^{4}-\frac{63}{536}a^{3}-\frac{32}{67}a^{2}-\frac{31}{134}a+\frac{75}{1072}$, $\frac{1}{174956832}a^{15}+\frac{6497}{174956832}a^{14}+\frac{1091}{7289868}a^{13}+\frac{2684021}{87478416}a^{12}+\frac{4808345}{58318944}a^{11}+\frac{6221521}{174956832}a^{10}-\frac{19644025}{174956832}a^{9}+\frac{15410653}{174956832}a^{8}+\frac{21712183}{174956832}a^{7}-\frac{6342079}{174956832}a^{6}-\frac{7710529}{174956832}a^{5}-\frac{5160293}{58318944}a^{4}+\frac{26190709}{87478416}a^{3}+\frac{1034923}{4859912}a^{2}-\frac{13159199}{58318944}a-\frac{104041}{19439648}$ 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:  $2$, $3$

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:   \( \frac{909313}{19439648} a^{15} - \frac{3834697}{58318944} a^{14} - \frac{116787}{1214978} a^{13} + \frac{7819993}{9719824} a^{12} - \frac{44980343}{58318944} a^{11} - \frac{48183353}{58318944} a^{10} + \frac{155472245}{58318944} a^{9} - \frac{142707677}{58318944} a^{8} + \frac{16766399}{19439648} a^{7} - \frac{228814249}{58318944} a^{6} + \frac{636283309}{58318944} a^{5} - \frac{763800625}{58318944} a^{4} + \frac{62988187}{29159472} a^{3} + \frac{109789405}{14579736} a^{2} + \frac{987131}{19439648} a - \frac{8652165}{19439648} \)  (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:   $\frac{2099839}{87478416}a^{15}-\frac{2782069}{87478416}a^{14}-\frac{977941}{21869604}a^{13}+\frac{17596961}{43739208}a^{12}-\frac{32472127}{87478416}a^{11}-\frac{30347665}{87478416}a^{10}+\frac{109361501}{87478416}a^{9}-\frac{101307365}{87478416}a^{8}+\frac{54965117}{87478416}a^{7}-\frac{205040953}{87478416}a^{6}+\frac{508010117}{87478416}a^{5}-\frac{608311625}{87478416}a^{4}+\frac{108657473}{43739208}a^{3}+\frac{5892425}{3644934}a^{2}+\frac{43571639}{29159472}a-\frac{7927091}{9719824}$, $\frac{258625}{10934802}a^{15}-\frac{518315}{21869604}a^{14}-\frac{1326893}{21869604}a^{13}+\frac{2085338}{5467401}a^{12}-\frac{5111201}{21869604}a^{11}-\frac{11949335}{21869604}a^{10}+\frac{24326179}{21869604}a^{9}-\frac{17139769}{21869604}a^{8}+\frac{661801}{21869604}a^{7}-\frac{43855571}{21869604}a^{6}+\frac{103924795}{21869604}a^{5}-\frac{95256523}{21869604}a^{4}-\frac{29045227}{21869604}a^{3}+\frac{13650323}{3644934}a^{2}+\frac{5958155}{3644934}a-\frac{558443}{2429956}$, $\frac{2479577}{87478416}a^{15}-\frac{5330011}{87478416}a^{14}-\frac{86695}{7289868}a^{13}+\frac{107015}{217608}a^{12}-\frac{74101501}{87478416}a^{11}+\frac{14089613}{87478416}a^{10}+\frac{128509639}{87478416}a^{9}-\frac{242976263}{87478416}a^{8}+\frac{237859943}{87478416}a^{7}-\frac{377006747}{87478416}a^{6}+\frac{92522887}{9719824}a^{5}-\frac{1310727227}{87478416}a^{4}+\frac{542326301}{43739208}a^{3}-\frac{9842921}{2429956}a^{2}+\frac{67616237}{29159472}a-\frac{5562737}{9719824}$, $\frac{7765025}{174956832}a^{15}-\frac{65413}{870432}a^{14}-\frac{356158}{5467401}a^{13}+\frac{68007605}{87478416}a^{12}-\frac{168942361}{174956832}a^{11}-\frac{75260237}{174956832}a^{10}+\frac{456394855}{174956832}a^{9}-\frac{185031311}{58318944}a^{8}+\frac{347932315}{174956832}a^{7}-\frac{767984285}{174956832}a^{6}+\frac{2025315407}{174956832}a^{5}-\frac{2778626441}{174956832}a^{4}+\frac{662849947}{87478416}a^{3}+\frac{52549093}{14579736}a^{2}-\frac{4959599}{58318944}a-\frac{14004259}{19439648}$, $\frac{2871127}{174956832}a^{15}-\frac{7860343}{174956832}a^{14}+\frac{4933}{7289868}a^{13}+\frac{27093383}{87478416}a^{12}-\frac{36884885}{58318944}a^{11}+\frac{29122441}{174956832}a^{10}+\frac{184729433}{174956832}a^{9}-\frac{337388111}{174956832}a^{8}+\frac{338865133}{174956832}a^{7}-\frac{436845127}{174956832}a^{6}+\frac{1128001841}{174956832}a^{5}-\frac{604092769}{58318944}a^{4}+\frac{793850125}{87478416}a^{3}-\frac{12784843}{4859912}a^{2}+\frac{52266631}{58318944}a-\frac{9554529}{19439648}$, $\frac{443447}{174956832}a^{15}-\frac{888925}{174956832}a^{14}-\frac{22166}{5467401}a^{13}+\frac{1425253}{29159472}a^{12}-\frac{14554187}{174956832}a^{11}-\frac{639469}{19439648}a^{10}+\frac{41083201}{174956832}a^{9}-\frac{75035033}{174956832}a^{8}+\frac{6871603}{58318944}a^{7}+\frac{8273563}{174956832}a^{6}+\frac{14897059}{58318944}a^{5}-\frac{28247053}{174956832}a^{4}-\frac{2828553}{9719824}a^{3}+\frac{34486873}{14579736}a^{2}-\frac{47414523}{19439648}a+\frac{10697605}{19439648}$, $\frac{1249841}{174956832}a^{15}-\frac{913583}{174956832}a^{14}-\frac{173071}{10934802}a^{13}+\frac{8964661}{87478416}a^{12}-\frac{9265693}{174956832}a^{11}-\frac{5674669}{58318944}a^{10}+\frac{36592207}{174956832}a^{9}-\frac{53122219}{174956832}a^{8}+\frac{15509317}{58318944}a^{7}-\frac{118642151}{174956832}a^{6}+\frac{213859639}{174956832}a^{5}-\frac{3583373}{2611296}a^{4}+\frac{24439211}{29159472}a^{3}+\frac{2834695}{14579736}a^{2}+\frac{1133627}{19439648}a+\frac{4859343}{19439648}$ 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:  \( 8400.0296703 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 8400.0296703 \cdot 1}{6\cdot\sqrt{23595621172490797056}}\cr\approx \mathstrut & 0.70008828285 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9)
 
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 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9, 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 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9);
 
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 - 2*x^15 - x^14 + 18*x^13 - 27*x^12 - 4*x^11 + 62*x^10 - 88*x^9 + 62*x^8 - 112*x^7 + 294*x^6 - 444*x^5 + 277*x^4 + 42*x^3 - 33*x^2 - 18*x + 9);
 
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

\(\Q(\sqrt{-66}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{22})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), 4.2.8712.1 x2, 4.2.8712.2 x2, 4.0.2112.2 x2, 4.0.2112.1 x2, 8.0.4857532416.2, 8.0.4857532416.6, 8.0.4857532416.8, 8.0.75898944.1 x2, 8.0.539725824.3 x2, 8.4.4857532416.2 x2, 8.0.40144896.1 x2

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: 8.0.75898944.1, 8.0.539725824.3, 8.4.4857532416.2, 8.0.40144896.1
Minimal sibling: 8.0.40144896.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ R ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(2\) Copy content Toggle raw display 2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$$2$$2$$6$$C_2^2$$[3]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(11\) Copy content Toggle raw display 11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$