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Properties

Label 16.0.162...856.13
Degree $16$
Signature $[0, 8]$
Discriminant $1.624\times 10^{20}$
Root discriminant \(18.33\)
Ramified primes $2,3,7$
Class number $4$ (GRH)
Class group [4] (GRH)
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 - 8*x^15 + 32*x^14 - 84*x^13 + 176*x^12 - 328*x^11 + 498*x^10 - 532*x^9 + 366*x^8 - 164*x^7 + 72*x^6 - 52*x^5 + 85*x^4 - 116*x^3 + 90*x^2 - 36*x + 22)
 
gp: K = bnfinit(y^16 - 8*y^15 + 32*y^14 - 84*y^13 + 176*y^12 - 328*y^11 + 498*y^10 - 532*y^9 + 366*y^8 - 164*y^7 + 72*y^6 - 52*y^5 + 85*y^4 - 116*y^3 + 90*y^2 - 36*y + 22, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 176*x^12 - 328*x^11 + 498*x^10 - 532*x^9 + 366*x^8 - 164*x^7 + 72*x^6 - 52*x^5 + 85*x^4 - 116*x^3 + 90*x^2 - 36*x + 22);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 176*x^12 - 328*x^11 + 498*x^10 - 532*x^9 + 366*x^8 - 164*x^7 + 72*x^6 - 52*x^5 + 85*x^4 - 116*x^3 + 90*x^2 - 36*x + 22)
 

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 176 x^{12} - 328 x^{11} + 498 x^{10} - 532 x^{9} + 366 x^{8} + \cdots + 22 \) 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:   \(162447943996702457856\) \(\medspace = 2^{32}\cdot 3^{8}\cdot 7^{8}\) 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:  \(18.33\)
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:  $2^{2}3^{1/2}7^{1/2}\approx 18.33030277982336$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{13}-\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{9}-\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{2}+\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{126973}a^{14}-\frac{1}{18139}a^{13}-\frac{430}{7469}a^{12}+\frac{43951}{126973}a^{11}+\frac{62232}{126973}a^{10}+\frac{491}{1309}a^{9}+\frac{62479}{126973}a^{8}-\frac{4721}{11543}a^{7}+\frac{55639}{126973}a^{6}-\frac{40335}{126973}a^{5}+\frac{4227}{18139}a^{4}-\frac{32884}{126973}a^{3}+\frac{55716}{126973}a^{2}+\frac{29179}{126973}a-\frac{394}{1649}$, $\frac{1}{131163109}a^{15}+\frac{509}{131163109}a^{14}+\frac{8169767}{131163109}a^{13}+\frac{8679067}{131163109}a^{12}+\frac{17135997}{131163109}a^{11}-\frac{8671550}{131163109}a^{10}+\frac{7206432}{131163109}a^{9}+\frac{31915148}{131163109}a^{8}+\frac{27386019}{131163109}a^{7}-\frac{63222785}{131163109}a^{6}+\frac{60370615}{131163109}a^{5}-\frac{40941443}{131163109}a^{4}-\frac{2909120}{131163109}a^{3}-\frac{39768646}{131163109}a^{2}-\frac{21542198}{131163109}a+\frac{5273461}{11923919}$ 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

$C_{4}$, which has order $4$ (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:   $\frac{1899202}{131163109}a^{15}-\frac{6812613}{131163109}a^{14}-\frac{1777807}{131163109}a^{13}+\frac{73553145}{131163109}a^{12}-\frac{33370979}{18737587}a^{11}+\frac{45717696}{11923919}a^{10}-\frac{1081022279}{131163109}a^{9}+\frac{1826519809}{131163109}a^{8}-\frac{1794883203}{131163109}a^{7}+\frac{752757820}{131163109}a^{6}+\frac{67820979}{131163109}a^{5}+\frac{48443678}{131163109}a^{4}-\frac{179943285}{131163109}a^{3}+\frac{250083655}{131163109}a^{2}-\frac{332087278}{131163109}a+\frac{21565815}{11923919}$, $\frac{9344906}{131163109}a^{15}-\frac{70086795}{131163109}a^{14}+\frac{252255099}{131163109}a^{13}-\frac{576675086}{131163109}a^{12}+\frac{1065555753}{131163109}a^{11}-\frac{168699402}{11923919}a^{10}+\frac{343971459}{18737587}a^{9}-\frac{1421333370}{131163109}a^{8}-\frac{393069315}{131163109}a^{7}+\frac{689941421}{131163109}a^{6}+\frac{192531462}{131163109}a^{5}-\frac{444383259}{131163109}a^{4}+\frac{430531554}{131163109}a^{3}-\frac{423384996}{131163109}a^{2}-\frac{144170655}{131163109}a+\frac{12765295}{11923919}$, $\frac{270188}{11923919}a^{15}-\frac{117714}{1703417}a^{14}-\frac{709697}{11923919}a^{13}+\frac{10446967}{11923919}a^{12}-\frac{29736886}{11923919}a^{11}+\frac{62384901}{11923919}a^{10}-\frac{134824386}{11923919}a^{9}+\frac{214401438}{11923919}a^{8}-\frac{186681865}{11923919}a^{7}+\frac{65433453}{11923919}a^{6}+\frac{7535882}{11923919}a^{5}+\frac{20541852}{11923919}a^{4}-\frac{44221566}{11923919}a^{3}+\frac{8972342}{1703417}a^{2}-\frac{34239721}{11923919}a+\frac{20862993}{11923919}$, $\frac{6551618}{131163109}a^{15}-\frac{43909122}{131163109}a^{14}+\frac{154956280}{131163109}a^{13}-\frac{54216556}{18737587}a^{12}+\frac{810936596}{131163109}a^{11}-\frac{1534404899}{131163109}a^{10}+\frac{2175118881}{131163109}a^{9}-\frac{2306274752}{131163109}a^{8}+\frac{2075512006}{131163109}a^{7}-\frac{1246526926}{131163109}a^{6}+\frac{100807702}{131163109}a^{5}+\frac{7154251}{7715477}a^{4}+\frac{640967505}{131163109}a^{3}-\frac{585027294}{131163109}a^{2}+\frac{345014622}{131163109}a-\frac{28327221}{11923919}$, $\frac{13260840}{131163109}a^{15}-\frac{7783100}{11923919}a^{14}+\frac{24759436}{11923919}a^{13}-\frac{557934858}{131163109}a^{12}+\frac{144285414}{18737587}a^{11}-\frac{1759740476}{131163109}a^{10}+\frac{175981832}{11923919}a^{9}-\frac{624550810}{131163109}a^{8}-\frac{417500304}{131163109}a^{7}-\frac{77189099}{131163109}a^{6}+\frac{464247914}{131163109}a^{5}-\frac{182051220}{131163109}a^{4}+\frac{5240295}{1352197}a^{3}-\frac{218300734}{131163109}a^{2}-\frac{253024458}{131163109}a-\frac{9598479}{11923919}$, $\frac{1623124}{131163109}a^{15}-\frac{12477132}{131163109}a^{14}+\frac{53926148}{131163109}a^{13}-\frac{161782399}{131163109}a^{12}+\frac{22878599}{7715477}a^{11}-\frac{782161846}{131163109}a^{10}+\frac{1311591767}{131163109}a^{9}-\frac{1823330997}{131163109}a^{8}+\frac{1954870310}{131163109}a^{7}-\frac{1474438495}{131163109}a^{6}+\frac{76472510}{11923919}a^{5}-\frac{49637594}{11923919}a^{4}+\frac{648379775}{131163109}a^{3}-\frac{95293906}{18737587}a^{2}+\frac{540859842}{131163109}a-\frac{9524351}{11923919}$, $\frac{715244}{131163109}a^{15}-\frac{5364330}{131163109}a^{14}+\frac{23024700}{131163109}a^{13}-\frac{68301545}{131163109}a^{12}+\frac{161352756}{131163109}a^{11}-\frac{28646634}{11923919}a^{10}+\frac{512702147}{131163109}a^{9}-\frac{687238254}{131163109}a^{8}+\frac{666208160}{131163109}a^{7}-\frac{364742969}{131163109}a^{6}+\frac{81779980}{131163109}a^{5}-\frac{40764051}{131163109}a^{4}+\frac{99335518}{131163109}a^{3}-\frac{3084351}{18737587}a^{2}+\frac{33595929}{131163109}a-\frac{3436357}{11923919}$ 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:  \( 3907.05547999 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3907.05547999 \cdot 4}{2\cdot\sqrt{162447943996702457856}}\cr\approx \mathstrut & 1.48922874790 \end{aligned}\] (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 + 176*x^12 - 328*x^11 + 498*x^10 - 532*x^9 + 366*x^8 - 164*x^7 + 72*x^6 - 52*x^5 + 85*x^4 - 116*x^3 + 90*x^2 - 36*x + 22)
 
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 + 176*x^12 - 328*x^11 + 498*x^10 - 532*x^9 + 366*x^8 - 164*x^7 + 72*x^6 - 52*x^5 + 85*x^4 - 116*x^3 + 90*x^2 - 36*x + 22, 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 + 176*x^12 - 328*x^11 + 498*x^10 - 532*x^9 + 366*x^8 - 164*x^7 + 72*x^6 - 52*x^5 + 85*x^4 - 116*x^3 + 90*x^2 - 36*x + 22);
 
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 + 176*x^12 - 328*x^11 + 498*x^10 - 532*x^9 + 366*x^8 - 164*x^7 + 72*x^6 - 52*x^5 + 85*x^4 - 116*x^3 + 90*x^2 - 36*x + 22);
 
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

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{3}) \), 4.0.3528.1, 4.0.14112.1, 4.0.392.1, 4.0.1568.1, \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{14})\), 8.0.12745506816.2, 8.0.3186376704.2, 8.0.39337984.3, 8.0.796594176.13, 8.0.12745506816.15, 8.0.796594176.8, 8.0.796594176.9

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: 16.0.67658452310163456.1, 16.0.634562281237118976.2, 16.0.10152996499793903616.4, 16.8.162447943996702457856.1, 16.0.162447943996702457856.11, 16.0.162447943996702457856.12, 16.0.162447943996702457856.15
Minimal sibling: 16.0.67658452310163456.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
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 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$