↑

Properties

Label 16.0.101...216.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.014\times 10^{20}$
Root discriminant \(17.80\)
Ramified primes $2,7$
Class number $2$
Class group [2]
Galois group $D_{8}$ (as 16T13)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 20*x^13 - 38*x^12 + 4*x^11 + 140*x^10 - 116*x^9 - 198*x^8 + 116*x^7 + 140*x^6 - 4*x^5 - 38*x^4 - 20*x^3 + 4*x^2 + 4*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 4*y^14 + 20*y^13 - 38*y^12 + 4*y^11 + 140*y^10 - 116*y^9 - 198*y^8 + 116*y^7 + 140*y^6 - 4*y^5 - 38*y^4 - 20*y^3 + 4*y^2 + 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 4*x^14 + 20*x^13 - 38*x^12 + 4*x^11 + 140*x^10 - 116*x^9 - 198*x^8 + 116*x^7 + 140*x^6 - 4*x^5 - 38*x^4 - 20*x^3 + 4*x^2 + 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 4*x^14 + 20*x^13 - 38*x^12 + 4*x^11 + 140*x^10 - 116*x^9 - 198*x^8 + 116*x^7 + 140*x^6 - 4*x^5 - 38*x^4 - 20*x^3 + 4*x^2 + 4*x + 1)
 

\( x^{16} - 4 x^{15} + 4 x^{14} + 20 x^{13} - 38 x^{12} + 4 x^{11} + 140 x^{10} - 116 x^{9} - 198 x^{8} + \cdots + 1 \) 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:   \(101415451701035401216\) \(\medspace = 2^{44}\cdot 7^{8}\) 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:  \(17.80\)
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:  $2^{11/4}7^{1/2}\approx 17.798422345016238$
Ramified primes:   \(2\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{5}-\frac{1}{4}a$, $\frac{1}{113032}a^{14}-\frac{1983}{56516}a^{13}+\frac{1849}{113032}a^{12}-\frac{5431}{56516}a^{11}+\frac{121}{113032}a^{10}+\frac{2298}{14129}a^{9}+\frac{11977}{113032}a^{8}+\frac{1322}{14129}a^{7}-\frac{40235}{113032}a^{6}+\frac{23321}{56516}a^{5}-\frac{28379}{113032}a^{4}-\frac{5431}{56516}a^{3}-\frac{30107}{113032}a^{2}-\frac{4028}{14129}a-\frac{28259}{113032}$, $\frac{1}{4182184}a^{15}-\frac{5}{2091092}a^{14}+\frac{80317}{4182184}a^{13}+\frac{63125}{2091092}a^{12}+\frac{5629}{113032}a^{11}-\frac{243089}{1045546}a^{10}+\frac{907245}{4182184}a^{9}+\frac{233859}{1045546}a^{8}+\frac{993869}{4182184}a^{7}-\frac{1004269}{2091092}a^{6}-\frac{828591}{4182184}a^{5}+\frac{26225}{56516}a^{4}+\frac{1647461}{4182184}a^{3}-\frac{310785}{1045546}a^{2}-\frac{1871335}{4182184}a+\frac{83785}{1045546}$ 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

$C_{2}$, 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:   \( -1 \)  (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:   $\frac{413879}{2091092}a^{15}-\frac{1128707}{1045546}a^{14}+\frac{4136889}{2091092}a^{13}+\frac{2688633}{1045546}a^{12}-\frac{720735}{56516}a^{11}+\frac{12527935}{1045546}a^{10}+\frac{49277103}{2091092}a^{9}-\frac{62395101}{1045546}a^{8}-\frac{4871641}{2091092}a^{7}+\frac{67605853}{1045546}a^{6}-\frac{4090607}{2091092}a^{5}-\frac{708431}{28258}a^{4}-\frac{14077683}{2091092}a^{3}-\frac{334163}{1045546}a^{2}+\frac{4422087}{2091092}a+\frac{128843}{1045546}$, $\frac{304665}{1045546}a^{15}-\frac{5718667}{4182184}a^{14}+\frac{2073521}{1045546}a^{13}+\frac{20712805}{4182184}a^{12}-\frac{211619}{14129}a^{11}+\frac{37432827}{4182184}a^{10}+\frac{20492449}{522773}a^{9}-\frac{257598421}{4182184}a^{8}-\frac{34580079}{1045546}a^{7}+\frac{294689545}{4182184}a^{6}+\frac{15561887}{1045546}a^{5}-\frac{2956739}{113032}a^{4}-\frac{1763627}{522773}a^{3}-\frac{2548841}{4182184}a^{2}+\frac{706638}{522773}a+\frac{1974855}{4182184}$, $\frac{128843}{1045546}a^{15}-\frac{1444623}{2091092}a^{14}+\frac{1644079}{1045546}a^{13}+\frac{1016831}{2091092}a^{12}-\frac{204991}{28258}a^{11}+\frac{27697939}{2091092}a^{10}+\frac{5510085}{1045546}a^{9}-\frac{79168679}{2091092}a^{8}+\frac{36884187}{1045546}a^{7}+\frac{34763217}{2091092}a^{6}-\frac{49567833}{1045546}a^{5}+\frac{82699}{56516}a^{4}+\frac{21315913}{1045546}a^{3}+\frac{8923963}{2091092}a^{2}+\frac{849535}{1045546}a-\frac{3391343}{2091092}$, $\frac{755721}{4182184}a^{15}-\frac{2252927}{4182184}a^{14}-\frac{239239}{4182184}a^{13}+\frac{18739181}{4182184}a^{12}-\frac{353275}{113032}a^{11}-\frac{31767883}{4182184}a^{10}+\frac{114000273}{4182184}a^{9}+\frac{29682465}{4182184}a^{8}-\frac{276857691}{4182184}a^{7}-\frac{59465947}{4182184}a^{6}+\frac{278314237}{4182184}a^{5}+\frac{1967109}{113032}a^{4}-\frac{90891347}{4182184}a^{3}-\frac{31774175}{4182184}a^{2}+\frac{5280029}{4182184}a+\frac{4424077}{4182184}$, $\frac{1020525}{2091092}a^{15}-\frac{1050537}{522773}a^{14}+\frac{4980919}{2091092}a^{13}+\frac{18209533}{2091092}a^{12}-\frac{1056765}{56516}a^{11}+\frac{3978416}{522773}a^{10}+\frac{126693357}{2091092}a^{9}-\frac{127939913}{2091092}a^{8}-\frac{138225263}{2091092}a^{7}+\frac{22821723}{522773}a^{6}+\frac{77924411}{2091092}a^{5}+\frac{226065}{56516}a^{4}-\frac{11143725}{2091092}a^{3}-\frac{1458638}{522773}a^{2}+\frac{2948589}{2091092}a+\frac{533927}{2091092}$, $\frac{769957}{4182184}a^{15}-\frac{3262123}{4182184}a^{14}+\frac{3624761}{4182184}a^{13}+\frac{15646223}{4182184}a^{12}-\frac{940291}{113032}a^{11}+\frac{8199333}{4182184}a^{10}+\frac{117346101}{4182184}a^{9}-\frac{127224933}{4182184}a^{8}-\frac{147129583}{4182184}a^{7}+\frac{172513297}{4182184}a^{6}+\frac{75805917}{4182184}a^{5}-\frac{1680377}{113032}a^{4}-\frac{16659755}{4182184}a^{3}+\frac{2257145}{4182184}a^{2}-\frac{2780991}{4182184}a+\frac{3426463}{4182184}$, $\frac{762839}{2091092}a^{15}-\frac{2757525}{2091092}a^{14}+\frac{1692761}{2091092}a^{13}+\frac{8596351}{1045546}a^{12}-\frac{646783}{56516}a^{11}-\frac{11784275}{2091092}a^{10}+\frac{115673187}{2091092}a^{9}-\frac{24385617}{1045546}a^{8}-\frac{211993637}{2091092}a^{7}+\frac{56523675}{2091092}a^{6}+\frac{177060077}{2091092}a^{5}+\frac{71683}{28258}a^{4}-\frac{53775551}{2091092}a^{3}-\frac{14758515}{2091092}a^{2}+\frac{1249519}{2091092}a+\frac{917089}{1045546}$ 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:  \( 3094.8131536 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3094.8131536 \cdot 2}{2\cdot\sqrt{101415451701035401216}}\cr\approx \mathstrut & 0.74648539683 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 20*x^13 - 38*x^12 + 4*x^11 + 140*x^10 - 116*x^9 - 198*x^8 + 116*x^7 + 140*x^6 - 4*x^5 - 38*x^4 - 20*x^3 + 4*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 - 4*x^15 + 4*x^14 + 20*x^13 - 38*x^12 + 4*x^11 + 140*x^10 - 116*x^9 - 198*x^8 + 116*x^7 + 140*x^6 - 4*x^5 - 38*x^4 - 20*x^3 + 4*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 - 4*x^15 + 4*x^14 + 20*x^13 - 38*x^12 + 4*x^11 + 140*x^10 - 116*x^9 - 198*x^8 + 116*x^7 + 140*x^6 - 4*x^5 - 38*x^4 - 20*x^3 + 4*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 - 4*x^15 + 4*x^14 + 20*x^13 - 38*x^12 + 4*x^11 + 140*x^10 - 116*x^9 - 198*x^8 + 116*x^7 + 140*x^6 - 4*x^5 - 38*x^4 - 20*x^3 + 4*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

$D_8$ (as 16T13):

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 7 conjugacy class representatives for $D_{8}$
Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.0.1568.1 x2, 4.2.1792.1 x2, 8.0.157351936.3, 8.0.1258815488.2 x4, 8.2.1438646272.5 x4

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: 8.2.1438646272.5, 8.0.1258815488.2
Minimal sibling: 8.0.1258815488.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.8.22.84$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
2.8.22.84$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
\(7\) 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}$