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Properties

Label 16.0.5236602731044893.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.237\times 10^{15}$
Root discriminant \(9.60\)
Ramified primes $3,13,181,853$
Class number $1$
Class group trivial
Galois group $C_4^4.C_2\wr D_4$ (as 16T1823)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*x + 1)
 
gp: K = bnfinit(y^16 - 7*y^15 + 26*y^14 - 66*y^13 + 125*y^12 - 180*y^11 + 195*y^10 - 148*y^9 + 60*y^8 + 21*y^7 - 58*y^6 + 51*y^5 - 22*y^4 - 2*y^3 + 9*y^2 - 5*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*x + 1)
 

\( x^{16} - 7 x^{15} + 26 x^{14} - 66 x^{13} + 125 x^{12} - 180 x^{11} + 195 x^{10} - 148 x^{9} + 60 x^{8} + \cdots + 1 \) 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:   \(5236602731044893\) \(\medspace = 3^{8}\cdot 13^{4}\cdot 181^{2}\cdot 853\) 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:  \(9.60\)
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:  $3^{1/2}13^{1/2}181^{1/2}853^{1/2}\approx 2453.839236787936$
Ramified primes:   \(3\), \(13\), \(181\), \(853\) 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:  \(\Q(\sqrt{853}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2041}a^{15}-\frac{924}{2041}a^{14}+\frac{319}{2041}a^{13}-\frac{726}{2041}a^{12}+\frac{501}{2041}a^{11}-\frac{372}{2041}a^{10}+\frac{472}{2041}a^{9}-\frac{280}{2041}a^{8}-\frac{346}{2041}a^{7}+\frac{948}{2041}a^{6}+\frac{92}{2041}a^{5}-\frac{632}{2041}a^{4}-\frac{122}{2041}a^{3}-\frac{383}{2041}a^{2}+\frac{168}{2041}a-\frac{986}{2041}$ 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:   \( -\frac{917}{157} a^{15} + \frac{5945}{157} a^{14} - \frac{20599}{157} a^{13} + \frac{48889}{157} a^{12} - \frac{86228}{157} a^{11} + \frac{113631}{157} a^{10} - \frac{108933}{157} a^{9} + \frac{66162}{157} a^{8} - \frac{9749}{157} a^{7} - \frac{29366}{157} a^{6} + \frac{37311}{157} a^{5} - \frac{23964}{157} a^{4} + \frac{4643}{157} a^{3} + \frac{5654}{157} a^{2} - \frac{5063}{157} a + \frac{1412}{157} \)  (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:   $\frac{3262}{2041}a^{15}-\frac{24023}{2041}a^{14}+\frac{91513}{2041}a^{13}-\frac{235367}{2041}a^{12}+\frac{448441}{2041}a^{11}-\frac{646066}{2041}a^{10}+\frac{692649}{2041}a^{9}-\frac{509242}{2041}a^{8}+\frac{179629}{2041}a^{7}+\frac{106393}{2041}a^{6}-\frac{218310}{2041}a^{5}+\frac{173311}{2041}a^{4}-\frac{61199}{2041}a^{3}-\frac{20664}{2041}a^{2}+\frac{35725}{2041}a-\frac{11962}{2041}$, $\frac{5790}{2041}a^{15}-\frac{35196}{2041}a^{14}+\frac{118283}{2041}a^{13}-\frac{274615}{2041}a^{12}+\frac{476082}{2041}a^{11}-\frac{617007}{2041}a^{10}+\frac{585748}{2041}a^{9}-\frac{353739}{2041}a^{8}+\frac{62152}{2041}a^{7}+\frac{143541}{2041}a^{6}-\frac{181670}{2041}a^{5}+\frac{118611}{2041}a^{4}-\frac{20604}{2041}a^{3}-\frac{25536}{2041}a^{2}+\frac{23655}{2041}a-\frac{6386}{2041}$, $\frac{1343}{2041}a^{15}-\frac{6127}{2041}a^{14}+\frac{14094}{2041}a^{13}-\frac{17789}{2041}a^{12}+\frac{3395}{2041}a^{11}+\frac{41269}{2041}a^{10}-\frac{98823}{2041}a^{9}+\frac{130128}{2041}a^{8}-\frac{99339}{2041}a^{7}+\frac{34277}{2041}a^{6}+\frac{21506}{2041}a^{5}-\frac{36458}{2041}a^{4}+\frac{30049}{2041}a^{3}-\frac{4119}{2041}a^{2}-\frac{9091}{2041}a+\frac{6534}{2041}$, $\frac{6956}{2041}a^{15}-\frac{43096}{2041}a^{14}+\frac{145308}{2041}a^{13}-\frac{337387}{2041}a^{12}+\frac{582654}{2041}a^{11}-\frac{748691}{2041}a^{10}+\frac{695244}{2041}a^{9}-\frac{394479}{2041}a^{8}+\frac{26096}{2041}a^{7}+\frac{216163}{2041}a^{6}-\frac{247883}{2041}a^{5}+\frac{147074}{2041}a^{4}-\frac{17945}{2041}a^{3}-\frac{41463}{2041}a^{2}+\frac{31771}{2041}a-\frac{6979}{2041}$, $\frac{1180}{2041}a^{15}-\frac{6549}{2041}a^{14}+\frac{19245}{2041}a^{13}-\frac{38239}{2041}a^{12}+\frac{54397}{2041}a^{11}-\frac{51170}{2041}a^{10}+\frac{24259}{2041}a^{9}+\frac{10447}{2041}a^{8}-\frac{24572}{2041}a^{7}+\frac{14459}{2041}a^{6}+\frac{2428}{2041}a^{5}-\frac{11000}{2041}a^{4}+\frac{13197}{2041}a^{3}-\frac{4961}{2041}a^{2}-\frac{3819}{2041}a+\frac{3972}{2041}$, $\frac{7097}{2041}a^{15}-\frac{40715}{2041}a^{14}+\frac{131098}{2041}a^{13}-\frac{292801}{2041}a^{12}+\frac{487974}{2041}a^{11}-\frac{601125}{2041}a^{10}+\frac{533204}{2041}a^{9}-\frac{276802}{2041}a^{8}-\frac{239}{2041}a^{7}+\frac{174305}{2041}a^{6}-\frac{177763}{2041}a^{5}+\frac{100823}{2041}a^{4}-\frac{450}{2041}a^{3}-\frac{28113}{2041}a^{2}+\frac{20762}{2041}a-\frac{3135}{2041}$, $a^{15}-7a^{14}+25a^{13}-60a^{12}+106a^{11}-139a^{10}+130a^{9}-74a^{8}+4a^{7}+39a^{6}-44a^{5}+26a^{4}-3a^{3}-9a^{2}+5a$ 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:  \( 29.7678343854 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 29.7678343854 \cdot 1}{6\cdot\sqrt{5236602731044893}}\cr\approx \mathstrut & 0.166536764681 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*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 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*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 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*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 - 7*x^15 + 26*x^14 - 66*x^13 + 125*x^12 - 180*x^11 + 195*x^10 - 148*x^9 + 60*x^8 + 21*x^7 - 58*x^6 + 51*x^5 - 22*x^4 - 2*x^3 + 9*x^2 - 5*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

$C_4^4.C_2\wr D_4$ (as 16T1823):

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 32768
The 230 conjugacy class representatives for $C_4^4.C_2\wr D_4$
Character table for $C_4^4.C_2\wr D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.117.1, 8.0.2477709.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ R ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ R ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(3\) 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}$
\(13\) Copy content Toggle raw display 13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} + 12 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + 3 x^{2} + 12 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(181\) Copy content Toggle raw display $\Q_{181}$$x + 179$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 179$$1$$1$$0$Trivial$[\ ]$
181.2.1.1$x^{2} + 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} + 177 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} + 177 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.1$x^{2} + 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} + 177 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.0.1$x^{4} + 6 x^{2} + 105 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
\(853\) Copy content Toggle raw display $\Q_{853}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{853}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$