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Properties

Label 16.0.268...000.3
Degree $16$
Signature $[0, 8]$
Discriminant $2.687\times 10^{19}$
Root discriminant \(16.38\)
Ramified primes $2,3,5$
Class number $2$
Class group [2]
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 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*x + 1)
 
gp: K = bnfinit(y^16 - 6*y^15 + 17*y^14 - 30*y^13 + 35*y^12 - 24*y^11 + 36*y^10 - 240*y^9 + 824*y^8 - 1620*y^7 + 2244*y^6 - 2292*y^5 + 1595*y^4 - 630*y^3 + 143*y^2 - 18*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*x + 1)
 

\( x^{16} - 6 x^{15} + 17 x^{14} - 30 x^{13} + 35 x^{12} - 24 x^{11} + 36 x^{10} - 240 x^{9} + 824 x^{8} + \cdots + 1 \) 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:   \(26873856000000000000\) \(\medspace = 2^{24}\cdot 3^{8}\cdot 5^{12}\) 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:  \(16.38\)
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:  $2^{3/2}3^{1/2}5^{3/4}\approx 16.3807251762544$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{12}a^{12}-\frac{1}{6}a^{10}+\frac{1}{6}a^{6}-\frac{1}{2}a^{4}+\frac{1}{3}a^{2}+\frac{1}{12}$, $\frac{1}{12}a^{13}-\frac{1}{6}a^{11}+\frac{1}{6}a^{7}-\frac{1}{2}a^{5}+\frac{1}{3}a^{3}+\frac{1}{12}a$, $\frac{1}{50616}a^{14}+\frac{1433}{50616}a^{13}+\frac{101}{50616}a^{12}+\frac{2005}{25308}a^{11}-\frac{1285}{25308}a^{10}-\frac{463}{4218}a^{9}+\frac{3319}{25308}a^{8}-\frac{985}{25308}a^{7}+\frac{412}{6327}a^{6}-\frac{1189}{8436}a^{5}+\frac{5873}{25308}a^{4}-\frac{439}{12654}a^{3}-\frac{19567}{50616}a^{2}-\frac{3955}{50616}a+\frac{12823}{50616}$, $\frac{1}{96625944}a^{15}+\frac{11}{12078243}a^{14}+\frac{935}{211899}a^{13}+\frac{339195}{10736216}a^{12}+\frac{5800505}{24156486}a^{11}+\frac{4260565}{48312972}a^{10}+\frac{158647}{48312972}a^{9}-\frac{5566849}{24156486}a^{8}+\frac{2819645}{48312972}a^{7}-\frac{9867361}{48312972}a^{6}-\frac{88441}{326439}a^{5}+\frac{8031323}{48312972}a^{4}-\frac{1690997}{32208648}a^{3}-\frac{39841}{108813}a^{2}+\frac{12346459}{48312972}a+\frac{31542671}{96625944}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

$C_{2}$, which has order $2$

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:   \( \frac{993509}{652878} a^{15} - \frac{5755225}{652878} a^{14} + \frac{1744735}{72542} a^{13} - \frac{932855}{22908} a^{12} + \frac{14680940}{326439} a^{11} - \frac{17924615}{652878} a^{10} + \frac{16115495}{326439} a^{9} - \frac{231869225}{652878} a^{8} + \frac{385390030}{326439} a^{7} - \frac{1451143885}{652878} a^{6} + \frac{966762505}{326439} a^{5} - \frac{1885549915}{652878} a^{4} + \frac{133961985}{72542} a^{3} - \frac{64600495}{108813} a^{2} + \frac{68940295}{652878} a - \frac{10273687}{1305756} \)  (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:   $\frac{497695}{635697}a^{15}-\frac{6124217}{1271394}a^{14}+\frac{8870162}{635697}a^{13}-\frac{15954685}{635697}a^{12}+\frac{18971896}{635697}a^{11}-\frac{8925247}{423798}a^{10}+\frac{18427421}{635697}a^{9}-\frac{242853559}{1271394}a^{8}+\frac{426890234}{635697}a^{7}-\frac{572055275}{423798}a^{6}+\frac{1204167793}{635697}a^{5}-\frac{2489119769}{1271394}a^{4}+\frac{880665185}{635697}a^{3}-\frac{351526984}{635697}a^{2}+\frac{67681933}{635697}a-\frac{3581429}{423798}$, $\frac{118250477}{96625944}a^{15}-\frac{78279931}{10736216}a^{14}+\frac{1979324243}{96625944}a^{13}-\frac{432063398}{12078243}a^{12}+\frac{661951151}{16104324}a^{11}-\frac{658339903}{24156486}a^{10}+\frac{2051381141}{48312972}a^{9}-\frac{82389739}{282532}a^{8}+\frac{8017179301}{8052162}a^{7}-\frac{93604136941}{48312972}a^{6}+\frac{128215366675}{48312972}a^{5}-\frac{21516707399}{8052162}a^{4}+\frac{174955625897}{96625944}a^{3}-\frac{64261207399}{96625944}a^{2}+\frac{1354281301}{10736216}a-\frac{458022173}{48312972}$, $\frac{35450}{70633}a^{15}-\frac{145083349}{48312972}a^{14}+\frac{204491339}{24156486}a^{13}-\frac{358004293}{24156486}a^{12}+\frac{412130513}{24156486}a^{11}-\frac{136757128}{12078243}a^{10}+\frac{1894525}{108813}a^{9}-\frac{1448537462}{12078243}a^{8}+\frac{9930649189}{24156486}a^{7}-\frac{9691154291}{12078243}a^{6}+\frac{8858248003}{8052162}a^{5}-\frac{13383132937}{12078243}a^{4}+\frac{9065950576}{12078243}a^{3}-\frac{13316534507}{48312972}a^{2}+\frac{631343287}{12078243}a-\frac{94890641}{24156486}$, $\frac{33646453}{24156486}a^{15}-\frac{389448905}{48312972}a^{14}+\frac{14320967}{652878}a^{13}-\frac{1786591637}{48312972}a^{12}+\frac{489887429}{12078243}a^{11}-\frac{195941029}{8052162}a^{10}+\frac{1076316587}{24156486}a^{9}-\frac{7842904847}{24156486}a^{8}+\frac{13018175296}{12078243}a^{7}-\frac{16274592581}{8052162}a^{6}+\frac{32355791819}{12078243}a^{5}-\frac{62716357945}{24156486}a^{4}+\frac{19784927854}{12078243}a^{3}-\frac{24327403037}{48312972}a^{2}+\frac{1928948125}{24156486}a-\frac{29936149}{5368108}$, $\frac{89104279}{96625944}a^{15}-\frac{495088555}{96625944}a^{14}+\frac{1291426811}{96625944}a^{13}-\frac{259953386}{12078243}a^{12}+\frac{56401523}{2542788}a^{11}-\frac{46066129}{4026081}a^{10}+\frac{1318835557}{48312972}a^{9}-\frac{10090918585}{48312972}a^{8}+\frac{8046138349}{12078243}a^{7}-\frac{19158937121}{16104324}a^{6}+\frac{73149581243}{48312972}a^{5}-\frac{16841135567}{12078243}a^{4}+\frac{76785853127}{96625944}a^{3}-\frac{17239652095}{96625944}a^{2}+\frac{2778999409}{96625944}a-\frac{53768369}{16104324}$, $\frac{591849}{5368108}a^{15}-\frac{6221203}{12078243}a^{14}+\frac{50671801}{48312972}a^{13}-\frac{54482033}{48312972}a^{12}+\frac{7480853}{24156486}a^{11}+\frac{25596169}{24156486}a^{10}+\frac{16103453}{8052162}a^{9}-\frac{533941657}{24156486}a^{8}+\frac{1385860435}{24156486}a^{7}-\frac{850090757}{12078243}a^{6}+\frac{416148781}{8052162}a^{5}-\frac{43186343}{24156486}a^{4}-\frac{2781339323}{48312972}a^{3}+\frac{1589443565}{24156486}a^{2}-\frac{737720003}{48312972}a+\frac{93522737}{48312972}$, $\frac{29235257}{96625944}a^{15}-\frac{183120239}{96625944}a^{14}+\frac{537021277}{96625944}a^{13}-\frac{487225555}{48312972}a^{12}+\frac{584737009}{48312972}a^{11}-\frac{34879276}{4026081}a^{10}+\frac{550139555}{48312972}a^{9}-\frac{3611922449}{48312972}a^{8}+\frac{3219109778}{12078243}a^{7}-\frac{8721475103}{16104324}a^{6}+\frac{36906835027}{48312972}a^{5}-\frac{9607297519}{12078243}a^{4}+\frac{54839171665}{96625944}a^{3}-\frac{21993035531}{96625944}a^{2}+\frac{4243713311}{96625944}a-\frac{7795989}{2684054}$ 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:  \( 4471.3211085 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4471.3211085 \cdot 2}{6\cdot\sqrt{26873856000000000000}}\cr\approx \mathstrut & 0.69837479870 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*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 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*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 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*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 - 6*x^15 + 17*x^14 - 30*x^13 + 35*x^12 - 24*x^11 + 36*x^10 - 240*x^9 + 824*x^8 - 1620*x^7 + 2244*x^6 - 2292*x^5 + 1595*x^4 - 630*x^3 + 143*x^2 - 18*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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.9000.2 x2, 4.0.9000.1 x2, 4.2.24000.2 x2, 4.2.24000.1 x2, 8.0.207360000.1, 8.0.81000000.1 x2, 8.0.5184000000.13 x2, 8.0.5184000000.10, 8.0.5184000000.12, 8.0.5184000000.11 x2, 8.4.576000000.1 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: 8.4.576000000.1, 8.0.81000000.1, 8.0.5184000000.11, 8.0.5184000000.13
Minimal sibling: 8.0.81000000.1

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 R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$$2$$2$$6$$C_2^2$$[3]^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(5\) Copy content Toggle raw display 5.8.6.2$x^{8} + 10 x^{4} - 25$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 10 x^{4} - 25$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$