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Properties

Label 16.0.122444006400000000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.224\times 10^{17}$
Root discriminant \(11.69\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $(C_4\times C_8):D_4$ (as 16T513)

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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 + 9*x^14 - 10*x^13 + 32*x^12 - 21*x^11 + 64*x^10 - 17*x^9 + 81*x^8 + x^7 + 52*x^6 + 9*x^5 + 8*x^4 + 5*x^3 - 3*x^2 + x + 1)
 
gp: K = bnfinit(y^16 - 2*y^15 + 9*y^14 - 10*y^13 + 32*y^12 - 21*y^11 + 64*y^10 - 17*y^9 + 81*y^8 + y^7 + 52*y^6 + 9*y^5 + 8*y^4 + 5*y^3 - 3*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 9*x^14 - 10*x^13 + 32*x^12 - 21*x^11 + 64*x^10 - 17*x^9 + 81*x^8 + x^7 + 52*x^6 + 9*x^5 + 8*x^4 + 5*x^3 - 3*x^2 + x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 2*x^15 + 9*x^14 - 10*x^13 + 32*x^12 - 21*x^11 + 64*x^10 - 17*x^9 + 81*x^8 + x^7 + 52*x^6 + 9*x^5 + 8*x^4 + 5*x^3 - 3*x^2 + x + 1)
 

\( x^{16} - 2 x^{15} + 9 x^{14} - 10 x^{13} + 32 x^{12} - 21 x^{11} + 64 x^{10} - 17 x^{9} + 81 x^{8} + \cdots + 1 \) 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:   \(122444006400000000\) \(\medspace = 2^{16}\cdot 3^{14}\cdot 5^{8}\) 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:  \(11.69\)
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:  $2^{5/2}3^{7/8}5^{1/2}\approx 33.07814062698043$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{4}$, $\frac{1}{33}a^{14}-\frac{5}{33}a^{13}+\frac{1}{11}a^{12}-\frac{2}{33}a^{11}-\frac{1}{11}a^{10}-\frac{1}{11}a^{9}+\frac{4}{33}a^{8}-\frac{10}{33}a^{7}+\frac{16}{33}a^{6}+\frac{3}{11}a^{5}+\frac{8}{33}a^{4}+\frac{5}{33}a^{3}-\frac{10}{33}a^{2}-\frac{4}{33}a+\frac{10}{33}$, $\frac{1}{1023}a^{15}+\frac{2}{341}a^{14}-\frac{43}{341}a^{13}-\frac{112}{1023}a^{12}-\frac{58}{1023}a^{11}-\frac{113}{1023}a^{10}+\frac{59}{1023}a^{9}-\frac{10}{1023}a^{8}+\frac{280}{1023}a^{7}+\frac{3}{341}a^{6}+\frac{7}{33}a^{5}+\frac{71}{1023}a^{4}+\frac{37}{341}a^{3}+\frac{304}{1023}a^{2}+\frac{197}{1023}a-\frac{37}{93}$ 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:   \( \frac{10}{33} a^{15} - \frac{35}{33} a^{14} + \frac{40}{11} a^{13} - \frac{76}{11} a^{12} + \frac{457}{33} a^{11} - \frac{212}{11} a^{10} + \frac{908}{33} a^{9} - \frac{325}{11} a^{8} + \frac{337}{11} a^{7} - \frac{88}{3} a^{6} + \frac{556}{33} a^{5} - \frac{156}{11} a^{4} + \frac{21}{11} a^{3} - \frac{47}{33} a^{2} - \frac{16}{11} a + \frac{29}{33} \)  (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:   $\frac{41}{1023}a^{15}-\frac{188}{1023}a^{14}+\frac{632}{1023}a^{13}-\frac{487}{341}a^{12}+\frac{2923}{1023}a^{11}-\frac{5036}{1023}a^{10}+\frac{7472}{1023}a^{9}-\frac{9989}{1023}a^{8}+\frac{11387}{1023}a^{7}-\frac{4124}{341}a^{6}+\frac{116}{11}a^{5}-\frac{234}{31}a^{4}+\frac{5450}{1023}a^{3}-\frac{1951}{1023}a^{2}+\frac{202}{341}a+\frac{152}{341}$, $\frac{7}{1023}a^{15}+\frac{1}{93}a^{14}+\frac{25}{93}a^{13}-\frac{536}{1023}a^{12}+\frac{2384}{1023}a^{11}-\frac{2744}{1023}a^{10}+\frac{222}{31}a^{9}-\frac{5650}{1023}a^{8}+\frac{11818}{1023}a^{7}-\frac{1622}{341}a^{6}+\frac{105}{11}a^{5}-\frac{3161}{1023}a^{4}+\frac{281}{1023}a^{3}-\frac{290}{1023}a^{2}-\frac{2248}{1023}a+\frac{251}{1023}$, $\frac{424}{1023}a^{15}-\frac{742}{1023}a^{14}+\frac{1112}{341}a^{13}-\frac{2786}{1023}a^{12}+\frac{10283}{1023}a^{11}-\frac{3272}{1023}a^{10}+\frac{17483}{1023}a^{9}+\frac{3758}{1023}a^{8}+\frac{1721}{93}a^{7}+\frac{3752}{341}a^{6}+\frac{70}{11}a^{5}+\frac{8249}{1023}a^{4}-\frac{2443}{1023}a^{3}+\frac{2509}{1023}a^{2}-\frac{172}{1023}a-\frac{1169}{1023}$, $\frac{284}{1023}a^{15}-\frac{300}{341}a^{14}+\frac{3323}{1023}a^{13}-\frac{5861}{1023}a^{12}+\frac{1177}{93}a^{11}-\frac{16096}{1023}a^{10}+\frac{26614}{1023}a^{9}-\frac{23486}{1023}a^{8}+\frac{31904}{1023}a^{7}-\frac{6898}{341}a^{6}+\frac{61}{3}a^{5}-\frac{2496}{341}a^{4}+\frac{1735}{341}a^{3}+\frac{16}{31}a^{2}-\frac{131}{1023}a+\frac{569}{1023}$, $\frac{84}{341}a^{15}-\frac{116}{341}a^{14}+\frac{2026}{1023}a^{13}-\frac{1409}{1023}a^{12}+\frac{2506}{341}a^{11}-\frac{812}{341}a^{10}+\frac{17038}{1023}a^{9}-\frac{412}{1023}a^{8}+\frac{26416}{1023}a^{7}+\frac{3539}{1023}a^{6}+\frac{751}{33}a^{5}+\frac{663}{341}a^{4}+\frac{9806}{1023}a^{3}-\frac{613}{1023}a^{2}+\frac{478}{1023}a-\frac{150}{341}$, $\frac{10}{31}a^{15}-\frac{686}{1023}a^{14}+\frac{3155}{1023}a^{13}-\frac{3697}{1023}a^{12}+\frac{11767}{1023}a^{11}-\frac{8150}{1023}a^{10}+\frac{24740}{1023}a^{9}-\frac{7144}{1023}a^{8}+\frac{33469}{1023}a^{7}+\frac{893}{1023}a^{6}+\frac{854}{33}a^{5}+\frac{6535}{1023}a^{4}+\frac{3220}{341}a^{3}+\frac{1634}{341}a^{2}+\frac{995}{1023}a+\frac{664}{1023}$, $\frac{14}{33}a^{15}-\frac{38}{33}a^{14}+\frac{157}{33}a^{13}-\frac{251}{33}a^{12}+\frac{214}{11}a^{11}-\frac{710}{33}a^{10}+\frac{1423}{33}a^{9}-\frac{1057}{33}a^{8}+\frac{1906}{33}a^{7}-\frac{82}{3}a^{6}+\frac{1445}{33}a^{5}-\frac{115}{11}a^{4}+\frac{416}{33}a^{3}+\frac{20}{33}a^{2}-\frac{76}{33}a+\frac{15}{11}$ 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:  \( 230.49973438 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 230.49973438 \cdot 1}{6\cdot\sqrt{122444006400000000}}\cr\approx \mathstrut & 0.26667933289 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 9*x^14 - 10*x^13 + 32*x^12 - 21*x^11 + 64*x^10 - 17*x^9 + 81*x^8 + x^7 + 52*x^6 + 9*x^5 + 8*x^4 + 5*x^3 - 3*x^2 + 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 - 2*x^15 + 9*x^14 - 10*x^13 + 32*x^12 - 21*x^11 + 64*x^10 - 17*x^9 + 81*x^8 + x^7 + 52*x^6 + 9*x^5 + 8*x^4 + 5*x^3 - 3*x^2 + 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 - 2*x^15 + 9*x^14 - 10*x^13 + 32*x^12 - 21*x^11 + 64*x^10 - 17*x^9 + 81*x^8 + x^7 + 52*x^6 + 9*x^5 + 8*x^4 + 5*x^3 - 3*x^2 + 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 - 2*x^15 + 9*x^14 - 10*x^13 + 32*x^12 - 21*x^11 + 64*x^10 - 17*x^9 + 81*x^8 + x^7 + 52*x^6 + 9*x^5 + 8*x^4 + 5*x^3 - 3*x^2 + 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

$(C_4\times C_8):D_4$ (as 16T513):

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 256
The 46 conjugacy class representatives for $(C_4\times C_8):D_4$
Character table for $(C_4\times C_8):D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.7290000.1

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 R R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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 2.8.16.38$x^{8} - 4 x^{5} + 20 x^{4} + 24 x^{3} + 88 x^{2} + 56 x + 124$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
\(3\) Copy content Toggle raw display 3.16.14.3$x^{16} + 36$$8$$2$$14$16T49$[\ ]_{8}^{4}$
\(5\) Copy content Toggle raw display 5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$