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Properties

Label 16.0.110...616.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.109\times 10^{18}$
Root discriminant \(13.42\)
Ramified primes $2,3,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2\wr C_2$ (as 16T39)

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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 + 2*x^14 - 2*x^13 + 9*x^12 - 10*x^11 + 4*x^10 + 8*x^9 + 18*x^8 - 32*x^7 + 26*x^6 + 16*x^5 + 4*x^4 - 6*x^3 + 18*x^2 + 18*x + 9)
 
gp: K = bnfinit(y^16 - 2*y^15 + 2*y^14 - 2*y^13 + 9*y^12 - 10*y^11 + 4*y^10 + 8*y^9 + 18*y^8 - 32*y^7 + 26*y^6 + 16*y^5 + 4*y^4 - 6*y^3 + 18*y^2 + 18*y + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 2*x^14 - 2*x^13 + 9*x^12 - 10*x^11 + 4*x^10 + 8*x^9 + 18*x^8 - 32*x^7 + 26*x^6 + 16*x^5 + 4*x^4 - 6*x^3 + 18*x^2 + 18*x + 9);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 2*x^14 - 2*x^13 + 9*x^12 - 10*x^11 + 4*x^10 + 8*x^9 + 18*x^8 - 32*x^7 + 26*x^6 + 16*x^5 + 4*x^4 - 6*x^3 + 18*x^2 + 18*x + 9)
 

\( x^{16} - 2 x^{15} + 2 x^{14} - 2 x^{13} + 9 x^{12} - 10 x^{11} + 4 x^{10} + 8 x^{9} + 18 x^{8} + \cdots + 9 \) 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:   \(1108540930828271616\) \(\medspace = 2^{24}\cdot 3^{4}\cdot 13^{8}\) 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:  \(13.42\)
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:  $2^{3/2}3^{1/2}13^{1/2}\approx 17.663521732655695$
Ramified primes:   \(2\), \(3\), \(13\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}$, $\frac{1}{219}a^{14}-\frac{9}{73}a^{13}+\frac{12}{73}a^{12}-\frac{20}{219}a^{11}-\frac{10}{219}a^{10}-\frac{80}{219}a^{9}+\frac{92}{219}a^{8}-\frac{68}{219}a^{7}-\frac{95}{219}a^{6}-\frac{59}{219}a^{5}-\frac{19}{219}a^{4}-\frac{5}{73}a^{3}+\frac{25}{73}a^{2}+\frac{23}{73}a-\frac{23}{73}$, $\frac{1}{2255481}a^{15}+\frac{4583}{2255481}a^{14}+\frac{3097}{2255481}a^{13}+\frac{306392}{2255481}a^{12}+\frac{373895}{2255481}a^{11}+\frac{70912}{2255481}a^{10}+\frac{682127}{2255481}a^{9}-\frac{336389}{2255481}a^{8}+\frac{436354}{2255481}a^{7}+\frac{1029218}{2255481}a^{6}+\frac{343819}{2255481}a^{5}+\frac{405656}{2255481}a^{4}-\frac{39377}{751827}a^{3}-\frac{20146}{250609}a^{2}-\frac{52605}{250609}a-\frac{42838}{250609}$ 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:  $3$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:   \( \frac{1045}{30897} a^{15} + \frac{200}{30897} a^{14} + \frac{2479}{30897} a^{13} - \frac{5971}{30897} a^{12} + \frac{7112}{30897} a^{11} + \frac{1735}{30897} a^{10} + \frac{28925}{30897} a^{9} - \frac{11336}{30897} a^{8} + \frac{12004}{30897} a^{7} + \frac{8240}{30897} a^{6} + \frac{72034}{30897} a^{5} - \frac{27217}{30897} a^{4} + \frac{802}{3433} a^{3} + \frac{9490}{10299} a^{2} + \frac{10703}{3433} a + \frac{4043}{3433} \)  (order $4$) 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:   $\frac{71519}{2255481}a^{15}+\frac{139789}{2255481}a^{14}-\frac{232828}{2255481}a^{13}+\frac{17314}{2255481}a^{12}+\frac{77224}{2255481}a^{11}+\frac{1841681}{2255481}a^{10}-\frac{662951}{2255481}a^{9}+\frac{382697}{2255481}a^{8}+\frac{838703}{2255481}a^{7}+\frac{4160902}{2255481}a^{6}-\frac{1866787}{2255481}a^{5}+\frac{3494908}{2255481}a^{4}+\frac{106265}{250609}a^{3}+\frac{550740}{250609}a^{2}+\frac{396830}{250609}a+\frac{455022}{250609}$, $\frac{11590}{2255481}a^{15}-\frac{200953}{2255481}a^{14}+\frac{393577}{2255481}a^{13}-\frac{578365}{2255481}a^{12}+\frac{931823}{2255481}a^{11}-\frac{1998425}{2255481}a^{10}+\frac{2213948}{2255481}a^{9}-\frac{2358737}{2255481}a^{8}+\frac{863428}{2255481}a^{7}-\frac{2713684}{2255481}a^{6}+\frac{4803361}{2255481}a^{5}-\frac{7561720}{2255481}a^{4}+\frac{1424852}{751827}a^{3}-\frac{161429}{250609}a^{2}+\frac{12283}{250609}a-\frac{259136}{250609}$, $\frac{336797}{2255481}a^{15}-\frac{670811}{2255481}a^{14}+\frac{667622}{2255481}a^{13}-\frac{880886}{2255481}a^{12}+\frac{3137992}{2255481}a^{11}-\frac{3006886}{2255481}a^{10}+\frac{1510630}{2255481}a^{9}+\frac{1042802}{2255481}a^{8}+\frac{4804082}{2255481}a^{7}-\frac{8246456}{2255481}a^{6}+\frac{9660044}{2255481}a^{5}+\frac{186841}{2255481}a^{4}-\frac{33169}{751827}a^{3}+\frac{1158230}{751827}a^{2}+\frac{414124}{250609}a+\frac{35626}{250609}$, $\frac{52679}{751827}a^{15}-\frac{166085}{751827}a^{14}+\frac{185786}{751827}a^{13}-\frac{45044}{751827}a^{12}+\frac{146296}{250609}a^{11}-\frac{938282}{751827}a^{10}+\frac{81610}{250609}a^{9}+\frac{433153}{250609}a^{8}+\frac{299338}{751827}a^{7}-\frac{2935948}{751827}a^{6}+\frac{2688028}{751827}a^{5}+\frac{2508929}{751827}a^{4}-\frac{767787}{250609}a^{3}-\frac{243730}{751827}a^{2}+\frac{509881}{250609}a+\frac{371047}{250609}$, $\frac{460600}{2255481}a^{15}-\frac{746263}{2255481}a^{14}+\frac{715537}{2255481}a^{13}-\frac{1393960}{2255481}a^{12}+\frac{4439396}{2255481}a^{11}-\frac{3116615}{2255481}a^{10}+\frac{2262002}{2255481}a^{9}-\frac{1103447}{2255481}a^{8}+\frac{8792314}{2255481}a^{7}-\frac{8228302}{2255481}a^{6}+\frac{10097764}{2255481}a^{5}-\frac{3557887}{2255481}a^{4}+\frac{2256365}{751827}a^{3}+\frac{264689}{250609}a^{2}+\frac{594300}{250609}a+\frac{190680}{250609}$, $\frac{25577}{250609}a^{15}-\frac{69842}{751827}a^{14}+\frac{137534}{751827}a^{13}-\frac{305162}{751827}a^{12}+\frac{195677}{250609}a^{11}-\frac{30367}{250609}a^{10}+\frac{961085}{751827}a^{9}-\frac{333704}{751827}a^{8}+\frac{650261}{751827}a^{7}-\frac{7307}{250609}a^{6}+\frac{1166272}{250609}a^{5}-\frac{470018}{250609}a^{4}-\frac{183892}{751827}a^{3}+\frac{43354}{10299}a^{2}+\frac{1040726}{250609}a+\frac{291632}{250609}$, $\frac{88204}{751827}a^{15}-\frac{151303}{751827}a^{14}+\frac{7411}{751827}a^{13}-\frac{94415}{751827}a^{12}+\frac{265165}{250609}a^{11}-\frac{403066}{751827}a^{10}-\frac{778258}{751827}a^{9}+\frac{254674}{751827}a^{8}+\frac{1855714}{751827}a^{7}-\frac{1749499}{751827}a^{6}-\frac{1192430}{751827}a^{5}+\frac{342676}{250609}a^{4}+\frac{1310003}{751827}a^{3}-\frac{573166}{751827}a^{2}-\frac{462161}{250609}a-\frac{280198}{250609}$ Copy content Toggle raw display (assuming GRH)
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:  \( 533.004017987 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 533.004017987 \cdot 1}{4\cdot\sqrt{1108540930828271616}}\cr\approx \mathstrut & 0.307421028940 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 2*x^14 - 2*x^13 + 9*x^12 - 10*x^11 + 4*x^10 + 8*x^9 + 18*x^8 - 32*x^7 + 26*x^6 + 16*x^5 + 4*x^4 - 6*x^3 + 18*x^2 + 18*x + 9)
 
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 + 2*x^14 - 2*x^13 + 9*x^12 - 10*x^11 + 4*x^10 + 8*x^9 + 18*x^8 - 32*x^7 + 26*x^6 + 16*x^5 + 4*x^4 - 6*x^3 + 18*x^2 + 18*x + 9, 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 + 2*x^14 - 2*x^13 + 9*x^12 - 10*x^11 + 4*x^10 + 8*x^9 + 18*x^8 - 32*x^7 + 26*x^6 + 16*x^5 + 4*x^4 - 6*x^3 + 18*x^2 + 18*x + 9);
 
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 + 2*x^14 - 2*x^13 + 9*x^12 - 10*x^11 + 4*x^10 + 8*x^9 + 18*x^8 - 32*x^7 + 26*x^6 + 16*x^5 + 4*x^4 - 6*x^3 + 18*x^2 + 18*x + 9);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), 4.0.832.1 x2, 4.2.2704.1 x2, 4.4.8112.1, 4.0.8112.1, 4.2.8112.1, 4.2.507.1, \(\Q(i, \sqrt{13})\), 8.0.65804544.2, 8.4.1052872704.1, 8.0.1052872704.1, 8.0.65804544.1, 8.4.65804544.1, 8.0.116985856.1, 8.0.1052872704.3

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: 8.4.1052872704.1, 8.4.9475854336.1, 8.4.65804544.1, 8.0.65804544.1, 8.0.9475854336.5, 8.0.1052872704.1, 8.4.592240896.1, 8.0.592240896.2
Degree 16 siblings: 16.0.89791815397090000896.2, 16.0.350749278894882816.3, 16.8.89791815397090000896.1, 16.0.89791815397090000896.9, 16.0.531312517142544384.3, 16.0.89791815397090000896.3
Minimal sibling: 8.4.65804544.1

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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 Deg $16$$4$$4$$24$
\(3\) Copy content Toggle raw display 3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{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}$
\(13\) Copy content Toggle raw display 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}$
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}$