↑

Properties

Label 16.0.87500953582971201.1
Degree $16$
Signature $[0, 8]$
Discriminant $8.750\times 10^{16}$
Root discriminant \(11.45\)
Ramified primes $3,7,13$
Class number $1$
Class group trivial
Galois group $C_2^2\wr C_2$ (as 16T39)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^13 + 7*x^12 + 21*x^10 - 24*x^9 + 9*x^8 - 27*x^7 + 24*x^6 - 9*x^5 + 16*x^4 - 9*x^3 + 3*x^2 - 3*x + 1)
 
gp: K = bnfinit(y^16 - 9*y^13 + 7*y^12 + 21*y^10 - 24*y^9 + 9*y^8 - 27*y^7 + 24*y^6 - 9*y^5 + 16*y^4 - 9*y^3 + 3*y^2 - 3*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 9*x^13 + 7*x^12 + 21*x^10 - 24*x^9 + 9*x^8 - 27*x^7 + 24*x^6 - 9*x^5 + 16*x^4 - 9*x^3 + 3*x^2 - 3*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 9*x^13 + 7*x^12 + 21*x^10 - 24*x^9 + 9*x^8 - 27*x^7 + 24*x^6 - 9*x^5 + 16*x^4 - 9*x^3 + 3*x^2 - 3*x + 1)
 

\( x^{16} - 9 x^{13} + 7 x^{12} + 21 x^{10} - 24 x^{9} + 9 x^{8} - 27 x^{7} + 24 x^{6} - 9 x^{5} + 16 x^{4} + \cdots + 1 \) 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:   \(87500953582971201\) \(\medspace = 3^{12}\cdot 7^{8}\cdot 13^{4}\) 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:  \(11.45\)
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:  $3^{3/4}7^{1/2}13^{1/2}\approx 21.745111415357325$
Ramified primes:   \(3\), \(7\), \(13\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{1108}a^{15}+\frac{71}{1108}a^{14}-\frac{111}{554}a^{13}+\frac{9}{554}a^{12}+\frac{177}{1108}a^{11}-\frac{175}{1108}a^{10}-\frac{493}{1108}a^{9}+\frac{38}{277}a^{8}-\frac{1}{554}a^{7}+\frac{385}{1108}a^{6}+\frac{213}{1108}a^{5}+\frac{433}{1108}a^{4}-\frac{271}{554}a^{3}+\frac{3}{277}a^{2}-\frac{253}{1108}a-\frac{119}{554}$ 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:  $2$

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:   \( \frac{580}{277} a^{15} + \frac{184}{277} a^{14} + \frac{322}{277} a^{13} - \frac{5072}{277} a^{12} + \frac{2386}{277} a^{11} - \frac{1503}{277} a^{10} + \frac{13220}{277} a^{9} - \frac{8790}{277} a^{8} + \frac{6596}{277} a^{7} - \frac{18798}{277} a^{6} + \frac{8308}{277} a^{5} - \frac{5916}{277} a^{4} + \frac{10838}{277} a^{3} - \frac{1904}{277} a^{2} + \frac{2286}{277} a - \frac{1479}{277} \)  (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:   $\frac{577}{1108}a^{15}+\frac{62}{277}a^{14}+\frac{711}{1108}a^{13}-\frac{2563}{554}a^{12}+\frac{1855}{1108}a^{11}-\frac{2151}{554}a^{10}+\frac{8319}{554}a^{9}-\frac{8415}{1108}a^{8}+\frac{2897}{277}a^{7}-\frac{27709}{1108}a^{6}+\frac{6189}{554}a^{5}-\frac{3258}{277}a^{4}+\frac{18281}{1108}a^{3}-\frac{1039}{277}a^{2}+\frac{4707}{1108}a-\frac{3535}{1108}$, $\frac{1193}{1108}a^{15}+\frac{1049}{1108}a^{14}-\frac{17}{554}a^{13}-\frac{5329}{554}a^{12}-\frac{467}{1108}a^{11}+\frac{7839}{1108}a^{10}+\frac{24021}{1108}a^{9}-\frac{2864}{277}a^{8}-\frac{6179}{554}a^{7}-\frac{18243}{1108}a^{6}+\frac{8687}{1108}a^{5}+\frac{5227}{1108}a^{4}+\frac{3557}{554}a^{3}-\frac{22}{277}a^{2}-\frac{453}{1108}a+\frac{67}{277}$, $\frac{122}{277}a^{15}+\frac{577}{1108}a^{14}+\frac{525}{1108}a^{13}-\frac{1128}{277}a^{12}-\frac{855}{554}a^{11}-\frac{361}{1108}a^{10}+\frac{15087}{1108}a^{9}-\frac{891}{1108}a^{8}-\frac{211}{554}a^{7}-\frac{10489}{554}a^{6}+\frac{5055}{1108}a^{5}-\frac{2817}{1108}a^{4}+\frac{12227}{1108}a^{3}-\frac{198}{277}a^{2}+\frac{712}{277}a-\frac{1743}{1108}$, $\frac{1391}{1108}a^{15}-\frac{405}{1108}a^{14}+\frac{221}{277}a^{13}-\frac{3020}{277}a^{12}+\frac{12973}{1108}a^{11}-\frac{10745}{1108}a^{10}+\frac{31667}{1108}a^{9}-\frac{17549}{554}a^{8}+\frac{9415}{277}a^{7}-\frac{54475}{1108}a^{6}+\frac{34795}{1108}a^{5}-\frac{32027}{1108}a^{4}+\frac{8328}{277}a^{3}-\frac{5227}{554}a^{2}+\frac{10947}{1108}a-\frac{2653}{554}$, $\frac{859}{554}a^{15}+\frac{1157}{554}a^{14}+\frac{155}{554}a^{13}-\frac{3903}{277}a^{12}-\frac{4185}{554}a^{11}+\frac{3367}{277}a^{10}+\frac{19713}{554}a^{9}+\frac{1209}{554}a^{8}-\frac{7784}{277}a^{7}-\frac{17197}{554}a^{6}-\frac{1727}{277}a^{5}+\frac{10739}{554}a^{4}+\frac{8923}{554}a^{3}+\frac{1830}{277}a^{2}-\frac{1267}{554}a-\frac{1401}{554}$, $\frac{1095}{1108}a^{15}-\frac{923}{1108}a^{14}-\frac{248}{277}a^{13}-\frac{4549}{554}a^{12}+\frac{15981}{1108}a^{11}+\frac{2275}{1108}a^{10}+\frac{9733}{1108}a^{9}-\frac{20101}{554}a^{8}+\frac{6661}{554}a^{7}-\frac{3343}{1108}a^{6}+\frac{26039}{1108}a^{5}-\frac{3413}{1108}a^{4}-\frac{1424}{277}a^{3}-\frac{1424}{277}a^{2}-\frac{1697}{1108}a+\frac{993}{554}$, $\frac{222}{277}a^{15}+\frac{223}{554}a^{14}-\frac{233}{554}a^{13}-\frac{2098}{277}a^{12}+\frac{514}{277}a^{11}+\frac{3461}{554}a^{10}+\frac{9079}{554}a^{9}-\frac{7025}{554}a^{8}-\frac{2660}{277}a^{7}-\frac{3447}{277}a^{6}+\frac{5101}{554}a^{5}+\frac{3615}{554}a^{4}+\frac{2835}{554}a^{3}+\frac{171}{277}a^{2}-\frac{766}{277}a-\frac{135}{554}$ 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:  \( 168.672908686 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 168.672908686 \cdot 1}{6\cdot\sqrt{87500953582971201}}\cr\approx \mathstrut & 0.230848277145 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^13 + 7*x^12 + 21*x^10 - 24*x^9 + 9*x^8 - 27*x^7 + 24*x^6 - 9*x^5 + 16*x^4 - 9*x^3 + 3*x^2 - 3*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 - 9*x^13 + 7*x^12 + 21*x^10 - 24*x^9 + 9*x^8 - 27*x^7 + 24*x^6 - 9*x^5 + 16*x^4 - 9*x^3 + 3*x^2 - 3*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 - 9*x^13 + 7*x^12 + 21*x^10 - 24*x^9 + 9*x^8 - 27*x^7 + 24*x^6 - 9*x^5 + 16*x^4 - 9*x^3 + 3*x^2 - 3*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 - 9*x^13 + 7*x^12 + 21*x^10 - 24*x^9 + 9*x^8 - 27*x^7 + 24*x^6 - 9*x^5 + 16*x^4 - 9*x^3 + 3*x^2 - 3*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

$C_2^2\wr C_2$ (as 16T39):

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 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 4.0.189.1 x2, 4.0.5733.1, 4.2.1323.1 x2, 4.0.117.1, 4.0.2457.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.0.2457.2, 8.0.295805601.3, 8.0.295805601.2, 8.0.6036849.2, 8.0.1750329.1, 8.0.32867289.1, 8.0.295805601.1, 8.0.6036849.1

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

Galois closure: deg 32
Degree 8 siblings: 8.0.6036849.2, 8.0.6036849.1, 8.0.295805601.1, 8.0.49991146569.4, 8.0.1020227481.2, 8.0.295805601.2, 8.0.49991146569.6, 8.0.1020227481.1
Degree 16 siblings: 16.0.1040864112987605361.1, 16.0.2499114735283240471761.1, 16.0.2499114735283240471761.5, 16.0.2499114735283240471761.2, 16.0.2499114735283240471761.3, 16.4.2499114735283240471761.1
Minimal sibling: 8.0.6036849.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ R ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 6 x^{5} + 6 x^{4} + 18 x^{2} + 18 x + 9$$4$$2$$6$$D_4$$[\ ]_{4}^{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}$
\(13\) 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}$
13.2.0.1$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}$
13.4.2.1$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}$