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Properties

Label 16.8.983...000.1
Degree $16$
Signature $[8, 4]$
Discriminant $9.834\times 10^{18}$
Root discriminant \(15.38\)
Ramified primes $2,5,7$
Class number $1$
Class group trivial
Galois group $C_4\times S_4$ (as 16T181)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^13 - 2*x^12 - 18*x^11 - 26*x^10 + 18*x^9 + 81*x^8 - 14*x^7 - 98*x^6 - 2*x^5 + 60*x^4 + 6*x^3 - 16*x^2 + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 14*y^13 - 2*y^12 - 18*y^11 - 26*y^10 + 18*y^9 + 81*y^8 - 14*y^7 - 98*y^6 - 2*y^5 + 60*y^4 + 6*y^3 - 16*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 14*x^13 - 2*x^12 - 18*x^11 - 26*x^10 + 18*x^9 + 81*x^8 - 14*x^7 - 98*x^6 - 2*x^5 + 60*x^4 + 6*x^3 - 16*x^2 + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 14*x^13 - 2*x^12 - 18*x^11 - 26*x^10 + 18*x^9 + 81*x^8 - 14*x^7 - 98*x^6 - 2*x^5 + 60*x^4 + 6*x^3 - 16*x^2 + 1)
 

\( x^{16} - 4 x^{15} + 14 x^{13} - 2 x^{12} - 18 x^{11} - 26 x^{10} + 18 x^{9} + 81 x^{8} - 14 x^{7} + \cdots + 1 \) 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:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(9834496000000000000\) \(\medspace = 2^{24}\cdot 5^{12}\cdot 7^{4}\) 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:  \(15.38\)
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:  $2^{19/12}5^{3/4}7^{1/2}\approx 26.50985484327206$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{379}a^{15}+\frac{129}{379}a^{14}+\frac{102}{379}a^{13}-\frac{64}{379}a^{12}-\frac{176}{379}a^{11}+\frac{72}{379}a^{10}+\frac{75}{379}a^{9}+\frac{139}{379}a^{8}-\frac{3}{379}a^{7}-\frac{34}{379}a^{6}-\frac{72}{379}a^{5}-\frac{103}{379}a^{4}+\frac{5}{379}a^{3}-\frac{87}{379}a^{2}+\frac{162}{379}a-\frac{57}{379}$ 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:  $11$
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:   $\frac{1416}{379}a^{15}-\frac{3804}{379}a^{14}-\frac{4894}{379}a^{13}+\frac{13222}{379}a^{12}+\frac{13810}{379}a^{11}-\frac{6442}{379}a^{10}-\frac{43126}{379}a^{9}-\frac{31713}{379}a^{8}+\frac{69278}{379}a^{7}+\frac{66314}{379}a^{6}-\frac{46618}{379}a^{5}-\frac{53372}{379}a^{4}+\frac{10870}{379}a^{3}+\frac{14764}{379}a^{2}-\frac{2556}{379}a-\frac{743}{379}$, $\frac{2113}{379}a^{15}-\frac{5988}{379}a^{14}-\frac{6568}{379}a^{13}+\frac{20916}{379}a^{12}+\frac{18482}{379}a^{11}-\frac{13108}{379}a^{10}-\frac{65135}{379}a^{9}-\frac{38676}{379}a^{8}+\frac{113046}{379}a^{7}+\frac{89991}{379}a^{6}-\frac{85053}{379}a^{5}-\frac{78925}{379}a^{4}+\frac{26483}{379}a^{3}+\frac{24619}{379}a^{2}-\frac{5237}{379}a-\frac{1435}{379}$, $\frac{359}{379}a^{15}-\frac{685}{379}a^{14}-\frac{1661}{379}a^{13}+\frac{1659}{379}a^{12}+\frac{4657}{379}a^{11}+\frac{3487}{379}a^{10}-\frac{8322}{379}a^{9}-\frac{16045}{379}a^{8}+\frac{2713}{379}a^{7}+\frac{19630}{379}a^{6}+\frac{11294}{379}a^{5}-\frac{6278}{379}a^{4}-\frac{11470}{379}a^{3}-\frac{3566}{379}a^{2}+\frac{1687}{379}a+\frac{761}{379}$, $\frac{3877}{379}a^{15}-\frac{11138}{379}a^{14}-\frac{11592}{379}a^{13}+\frac{38396}{379}a^{12}+\frac{33200}{379}a^{11}-\frac{24056}{379}a^{10}-\frac{120440}{379}a^{9}-\frac{69013}{379}a^{8}+\frac{208189}{379}a^{7}+\frac{163423}{379}a^{6}-\frac{154074}{379}a^{5}-\frac{145780}{379}a^{4}+\frac{44778}{379}a^{3}+\frac{46249}{379}a^{2}-\frac{7130}{379}a-\frac{3064}{379}$, $\frac{258}{379}a^{15}-\frac{1586}{379}a^{14}+\frac{1302}{379}a^{13}+\frac{5849}{379}a^{12}-\frac{4476}{379}a^{11}-\frac{11365}{379}a^{10}-\frac{7938}{379}a^{9}+\frac{21081}{379}a^{8}+\frac{40158}{379}a^{7}-\frac{20521}{379}a^{6}-\frac{57234}{379}a^{5}-\frac{1181}{379}a^{4}+\frac{35400}{379}a^{3}+\frac{9769}{379}a^{2}-\frac{5579}{379}a-\frac{1062}{379}$, $\frac{1558}{379}a^{15}-\frac{3678}{379}a^{14}-\frac{6707}{379}a^{13}+\frac{12472}{379}a^{12}+\frac{20275}{379}a^{11}-\frac{1145}{379}a^{10}-\frac{50289}{379}a^{9}-\frac{52149}{379}a^{8}+\frac{62030}{379}a^{7}+\frac{100523}{379}a^{6}-\frac{20458}{379}a^{5}-\frac{75578}{379}a^{4}-\frac{13055}{379}a^{3}+\frac{17191}{379}a^{2}+\frac{3772}{379}a-\frac{499}{379}$, $\frac{450}{379}a^{15}-\frac{2590}{379}a^{14}+\frac{1936}{379}a^{13}+\frac{9100}{379}a^{12}-\frac{6432}{379}a^{11}-\frac{16491}{379}a^{10}-\frac{13246}{379}a^{9}+\frac{29956}{379}a^{8}+\frac{60048}{379}a^{7}-\frac{29702}{379}a^{6}-\frac{82428}{379}a^{5}+\frac{1025}{379}a^{4}+\frac{50004}{379}a^{3}+\frac{10878}{379}a^{2}-\frac{7827}{379}a-\frac{1394}{379}$, $\frac{2853}{379}a^{15}-\frac{7173}{379}a^{14}-\frac{11436}{379}a^{13}+\frac{25479}{379}a^{12}+\frac{33399}{379}a^{11}-\frac{9477}{379}a^{10}-\frac{91878}{379}a^{9}-\frac{80215}{379}a^{8}+\frac{133945}{379}a^{7}+\frac{165266}{379}a^{6}-\frac{75040}{379}a^{5}-\frac{133921}{379}a^{4}+\frac{3274}{379}a^{3}+\frac{38692}{379}a^{2}+\frac{943}{379}a-\frac{3062}{379}$, $\frac{1822}{379}a^{15}-\frac{4869}{379}a^{14}-\frac{6688}{379}a^{13}+\frac{17558}{379}a^{12}+\frac{19670}{379}a^{11}-\frac{10183}{379}a^{10}-\frac{60430}{379}a^{9}-\frac{42362}{379}a^{8}+\frac{98001}{379}a^{7}+\frac{99506}{379}a^{6}-\frac{67891}{379}a^{5}-\frac{91779}{379}a^{4}+\frac{14037}{379}a^{3}+\frac{33639}{379}a^{2}-\frac{77}{379}a-\frac{3419}{379}$, $a^{15}-3a^{14}-3a^{13}+11a^{12}+9a^{11}-9a^{10}-35a^{9}-17a^{8}+64a^{7}+50a^{6}-48a^{5}-50a^{4}+10a^{3}+16a^{2}$, $\frac{1629}{379}a^{15}-\frac{4752}{379}a^{14}-\frac{5150}{379}a^{13}+\frac{17782}{379}a^{12}+\frac{14222}{379}a^{11}-\frac{14604}{379}a^{10}-\frac{53302}{379}a^{9}-\frac{25983}{379}a^{8}+\frac{101612}{379}a^{7}+\frac{72337}{379}a^{6}-\frac{86210}{379}a^{5}-\frac{72279}{379}a^{4}+\frac{30885}{379}a^{3}+\frac{28069}{379}a^{2}-\frac{4434}{379}a-\frac{1893}{379}$ 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:  \( 2954.91741119 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 2954.91741119 \cdot 1}{2\cdot\sqrt{9834496000000000000}}\cr\approx \mathstrut & 0.187974412476 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 14*x^13 - 2*x^12 - 18*x^11 - 26*x^10 + 18*x^9 + 81*x^8 - 14*x^7 - 98*x^6 - 2*x^5 + 60*x^4 + 6*x^3 - 16*x^2 + 1)
 
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 - 4*x^15 + 14*x^13 - 2*x^12 - 18*x^11 - 26*x^10 + 18*x^9 + 81*x^8 - 14*x^7 - 98*x^6 - 2*x^5 + 60*x^4 + 6*x^3 - 16*x^2 + 1, 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(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 14*x^13 - 2*x^12 - 18*x^11 - 26*x^10 + 18*x^9 + 81*x^8 - 14*x^7 - 98*x^6 - 2*x^5 + 60*x^4 + 6*x^3 - 16*x^2 + 1);
 
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 - 4*x^15 + 14*x^13 - 2*x^12 - 18*x^11 - 26*x^10 + 18*x^9 + 81*x^8 - 14*x^7 - 98*x^6 - 2*x^5 + 60*x^4 + 6*x^3 - 16*x^2 + 1);
 
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

$C_4\times S_4$ (as 16T181):

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 96
The 20 conjugacy class representatives for $C_4\times S_4$
Character table for $C_4\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.2.14000.1, 8.4.196000000.1

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 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Minimal sibling: 12.4.12293120000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R R ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(2\) Copy content Toggle raw display Deg $16$$8$$2$$24$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(7\) Copy content Toggle raw display 7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)