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Properties

Label 16.0.26843545600000000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2.684\times 10^{16}$
Root discriminant \(10.64\)
Ramified primes $2,5$
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 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 74*x^10 - 128*x^9 + 199*x^8 - 204*x^7 + 126*x^6 - 36*x^5 + 2*x^4 - 8*x^3 + 2*x^2 + 4*x + 1)
 
gp: K = bnfinit(y^16 - 8*y^15 + 28*y^14 - 56*y^13 + 72*y^12 - 68*y^11 + 74*y^10 - 128*y^9 + 199*y^8 - 204*y^7 + 126*y^6 - 36*y^5 + 2*y^4 - 8*y^3 + 2*y^2 + 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 74*x^10 - 128*x^9 + 199*x^8 - 204*x^7 + 126*x^6 - 36*x^5 + 2*x^4 - 8*x^3 + 2*x^2 + 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 74*x^10 - 128*x^9 + 199*x^8 - 204*x^7 + 126*x^6 - 36*x^5 + 2*x^4 - 8*x^3 + 2*x^2 + 4*x + 1)
 

\( x^{16} - 8 x^{15} + 28 x^{14} - 56 x^{13} + 72 x^{12} - 68 x^{11} + 74 x^{10} - 128 x^{9} + 199 x^{8} + \cdots + 1 \) 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:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(26843545600000000\) \(\medspace = 2^{36}\cdot 5^{8}\) 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:  \(10.64\)
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:  $2^{9/4}5^{1/2}\approx 10.636591793889977$
Ramified primes:   \(2\), \(5\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{57}a^{14}-\frac{7}{57}a^{13}-\frac{3}{19}a^{12}-\frac{26}{57}a^{11}-\frac{7}{57}a^{10}+\frac{7}{19}a^{9}-\frac{6}{19}a^{8}+\frac{1}{19}a^{7}-\frac{6}{19}a^{6}-\frac{8}{57}a^{5}-\frac{26}{57}a^{4}+\frac{26}{57}a^{3}-\frac{28}{57}a^{2}-\frac{6}{19}a+\frac{7}{57}$, $\frac{1}{22857}a^{15}+\frac{193}{22857}a^{14}+\frac{1044}{7619}a^{13}+\frac{1038}{7619}a^{12}+\frac{3350}{7619}a^{11}-\frac{8276}{22857}a^{10}+\frac{11174}{22857}a^{9}-\frac{1522}{7619}a^{8}-\frac{8500}{22857}a^{7}-\frac{3356}{7619}a^{6}+\frac{5518}{22857}a^{5}-\frac{2434}{7619}a^{4}+\frac{8782}{22857}a^{3}-\frac{830}{22857}a^{2}+\frac{1993}{22857}a-\frac{268}{7619}$ 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:   \( -\frac{1574}{7619} a^{15} + \frac{11805}{7619} a^{14} - \frac{37969}{7619} a^{13} + \frac{67756}{7619} a^{12} - \frac{77445}{7619} a^{11} + \frac{74525}{7619} a^{10} - \frac{5446}{401} a^{9} + \frac{180612}{7619} a^{8} - \frac{241767}{7619} a^{7} + \frac{216052}{7619} a^{6} - \frac{139621}{7619} a^{5} + \frac{80487}{7619} a^{4} - \frac{55736}{7619} a^{3} + \frac{35651}{7619} a^{2} - \frac{9984}{7619} a + \frac{341}{7619} \)  (order $8$) 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:   $\frac{7654}{22857}a^{15}-\frac{56603}{22857}a^{14}+\frac{187414}{22857}a^{13}-\frac{19189}{1203}a^{12}+\frac{475258}{22857}a^{11}-\frac{478531}{22857}a^{10}+\frac{561515}{22857}a^{9}-\frac{908878}{22857}a^{8}+\frac{1318129}{22857}a^{7}-\frac{1368113}{22857}a^{6}+\frac{951421}{22857}a^{5}-\frac{436822}{22857}a^{4}+\frac{56208}{7619}a^{3}-\frac{20243}{7619}a^{2}-\frac{1222}{22857}a+\frac{15970}{22857}$, $\frac{2247}{7619}a^{15}-\frac{13845}{7619}a^{14}+\frac{95600}{22857}a^{13}-\frac{75863}{22857}a^{12}-\frac{58441}{22857}a^{11}+\frac{153791}{22857}a^{10}-\frac{7855}{7619}a^{9}-\frac{198344}{22857}a^{8}+\frac{60139}{22857}a^{7}+\frac{441962}{22857}a^{6}-\frac{241777}{7619}a^{5}+\frac{510040}{22857}a^{4}-\frac{102824}{22857}a^{3}-\frac{33962}{22857}a^{2}-\frac{1957}{1203}a+\frac{1355}{22857}$, $\frac{822}{7619}a^{15}-\frac{15287}{22857}a^{14}+\frac{38347}{22857}a^{13}-\frac{44537}{22857}a^{12}+\frac{2906}{7619}a^{11}+\frac{43186}{22857}a^{10}-\frac{5892}{7619}a^{9}-\frac{93911}{22857}a^{8}+\frac{94708}{22857}a^{7}+\frac{90452}{22857}a^{6}-\frac{268415}{22857}a^{5}+\frac{243179}{22857}a^{4}-\frac{12429}{7619}a^{3}-\frac{961}{7619}a^{2}-\frac{64078}{22857}a+\frac{1163}{7619}$, $\frac{254}{22857}a^{15}+\frac{6115}{22857}a^{14}-\frac{54592}{22857}a^{13}+\frac{186649}{22857}a^{12}-\frac{113104}{7619}a^{11}+\frac{123220}{7619}a^{10}-\frac{310456}{22857}a^{9}+\frac{145138}{7619}a^{8}-\frac{832492}{22857}a^{7}+\frac{1133131}{22857}a^{6}-\frac{314893}{7619}a^{5}+\frac{162511}{7619}a^{4}-\frac{203041}{22857}a^{3}+\frac{33051}{7619}a^{2}-\frac{76}{1203}a-\frac{16949}{22857}$, $\frac{1644}{7619}a^{15}-\frac{12330}{7619}a^{14}+\frac{121606}{22857}a^{13}-\frac{229424}{22857}a^{12}+\frac{275680}{22857}a^{11}-\frac{75603}{7619}a^{10}+\frac{12301}{1203}a^{9}-\frac{163793}{7619}a^{8}+\frac{794926}{22857}a^{7}-\frac{732173}{22857}a^{6}+\frac{299255}{22857}a^{5}+\frac{89368}{22857}a^{4}-\frac{58534}{22857}a^{3}-\frac{28388}{7619}a^{2}+\frac{5936}{7619}a+\frac{161}{22857}$, $\frac{8252}{22857}a^{15}-\frac{58682}{22857}a^{14}+\frac{59352}{7619}a^{13}-\frac{294488}{22857}a^{12}+\frac{15691}{1203}a^{11}-\frac{247060}{22857}a^{10}+\frac{387670}{22857}a^{9}-\frac{762572}{22857}a^{8}+\frac{964523}{22857}a^{7}-\frac{710122}{22857}a^{6}+\frac{299731}{22857}a^{5}-\frac{39378}{7619}a^{4}+\frac{133075}{22857}a^{3}-\frac{48200}{22857}a^{2}-\frac{39275}{22857}a-\frac{14900}{22857}$, $\frac{2737}{7619}a^{15}-\frac{19124}{7619}a^{14}+\frac{167939}{22857}a^{13}-\frac{261361}{22857}a^{12}+\frac{240563}{22857}a^{11}-\frac{61879}{7619}a^{10}+\frac{336551}{22857}a^{9}-\frac{12472}{401}a^{8}+\frac{853076}{22857}a^{7}-\frac{531919}{22857}a^{6}+\frac{152089}{22857}a^{5}-\frac{66169}{22857}a^{4}+\frac{195667}{22857}a^{3}-\frac{42551}{7619}a^{2}-\frac{14398}{7619}a-\frac{24041}{22857}$ 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:  \( 120.028472464 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 120.028472464 \cdot 1}{8\cdot\sqrt{26843545600000000}}\cr\approx \mathstrut & 0.222440211677 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 74*x^10 - 128*x^9 + 199*x^8 - 204*x^7 + 126*x^6 - 36*x^5 + 2*x^4 - 8*x^3 + 2*x^2 + 4*x + 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 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 74*x^10 - 128*x^9 + 199*x^8 - 204*x^7 + 126*x^6 - 36*x^5 + 2*x^4 - 8*x^3 + 2*x^2 + 4*x + 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(QQ); K<a> := NumberField(x^16 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 74*x^10 - 128*x^9 + 199*x^8 - 204*x^7 + 126*x^6 - 36*x^5 + 2*x^4 - 8*x^3 + 2*x^2 + 4*x + 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 - 8*x^15 + 28*x^14 - 56*x^13 + 72*x^12 - 68*x^11 + 74*x^10 - 128*x^9 + 199*x^8 - 204*x^7 + 126*x^6 - 36*x^5 + 2*x^4 - 8*x^3 + 2*x^2 + 4*x + 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

$C_2\times D_4$ (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\zeta_{8})\), 4.2.400.1 x2, 4.2.1600.1 x2, 4.0.320.1 x2, 4.0.1280.1 x2, 8.0.40960000.1, 8.4.40960000.1 x2, 8.0.163840000.2 x2, 8.0.2560000.1, 8.0.40960000.2, 8.0.40960000.3 x2, 8.0.6553600.1 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: 8.4.40960000.1, 8.0.40960000.3, 8.0.163840000.2, 8.0.6553600.1
Minimal sibling: 8.0.6553600.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 Deg $16$$8$$2$$36$
\(5\) 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}$