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Properties

Label 16.0.108...296.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.083\times 10^{18}$
Root discriminant \(13.40\)
Ramified primes $2,3,7$
Class number $1$
Class group trivial
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 40*x^13 - 96*x^12 + 48*x^11 + 226*x^10 - 692*x^9 + 1143*x^8 - 1312*x^7 + 1078*x^6 - 624*x^5 + 258*x^4 - 88*x^3 + 30*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 40*y^13 - 96*y^12 + 48*y^11 + 226*y^10 - 692*y^9 + 1143*y^8 - 1312*y^7 + 1078*y^6 - 624*y^5 + 258*y^4 - 88*y^3 + 30*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 40*x^13 - 96*x^12 + 48*x^11 + 226*x^10 - 692*x^9 + 1143*x^8 - 1312*x^7 + 1078*x^6 - 624*x^5 + 258*x^4 - 88*x^3 + 30*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 40*x^13 - 96*x^12 + 48*x^11 + 226*x^10 - 692*x^9 + 1143*x^8 - 1312*x^7 + 1078*x^6 - 624*x^5 + 258*x^4 - 88*x^3 + 30*x^2 - 8*x + 1)
 

\( x^{16} - 4 x^{15} + 40 x^{13} - 96 x^{12} + 48 x^{11} + 226 x^{10} - 692 x^{9} + 1143 x^{8} - 1312 x^{7} + \cdots + 1 \) 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:   \(1082535236962615296\) \(\medspace = 2^{36}\cdot 3^{8}\cdot 7^{4}\) 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:  \(13.40\)
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:  $2^{9/4}3^{1/2}7^{1/2}\approx 21.798526485920096$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{903}a^{14}+\frac{19}{903}a^{13}-\frac{65}{903}a^{12}+\frac{48}{301}a^{11}+\frac{9}{301}a^{10}+\frac{19}{903}a^{9}+\frac{52}{903}a^{8}+\frac{99}{301}a^{7}-\frac{10}{129}a^{6}+\frac{290}{903}a^{5}+\frac{149}{301}a^{4}-\frac{4}{21}a^{3}+\frac{22}{301}a^{2}-\frac{419}{903}a+\frac{304}{903}$, $\frac{1}{304311}a^{15}-\frac{32}{304311}a^{14}+\frac{11006}{304311}a^{13}-\frac{4867}{101437}a^{12}-\frac{45544}{304311}a^{11}-\frac{1097}{43473}a^{10}-\frac{2711}{43473}a^{9}+\frac{12695}{304311}a^{8}-\frac{31471}{304311}a^{7}+\frac{27940}{304311}a^{6}-\frac{9832}{43473}a^{5}+\frac{90809}{304311}a^{4}+\frac{16491}{101437}a^{3}+\frac{6765}{101437}a^{2}+\frac{33010}{101437}a+\frac{91652}{304311}$ 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 $1$

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:   \( \frac{1311956}{304311} a^{15} - \frac{35279}{2359} a^{14} - \frac{2375216}{304311} a^{13} + \frac{51037337}{304311} a^{12} - \frac{2299180}{7077} a^{11} + \frac{4106705}{101437} a^{10} + \frac{298871593}{304311} a^{9} - \frac{748498885}{304311} a^{8} + \frac{53038941}{14491} a^{7} - \frac{386247650}{101437} a^{6} + \frac{280260278}{101437} a^{5} - \frac{414213122}{304311} a^{4} + \frac{146250607}{304311} a^{3} - \frac{48651902}{304311} a^{2} + \frac{5410295}{101437} a - \frac{409240}{43473} \)  (order $24$) 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:   $\frac{362239}{304311}a^{15}-\frac{1689914}{304311}a^{14}+\frac{100199}{43473}a^{13}+\frac{5132819}{101437}a^{12}-\frac{43868896}{304311}a^{11}+\frac{30159245}{304311}a^{10}+\frac{89712169}{304311}a^{9}-\frac{14535727}{14491}a^{8}+\frac{172891996}{101437}a^{7}-\frac{604080146}{304311}a^{6}+\frac{497210177}{304311}a^{5}-\frac{277089038}{304311}a^{4}+\frac{4898023}{14491}a^{3}-\frac{10667624}{101437}a^{2}+\frac{12466684}{304311}a-\frac{950533}{101437}$, $\frac{7891}{101437}a^{15}-\frac{48488}{304311}a^{14}-\frac{167261}{304311}a^{13}+\frac{823597}{304311}a^{12}-\frac{337697}{304311}a^{11}-\frac{756258}{101437}a^{10}+\frac{4263097}{304311}a^{9}-\frac{1481153}{101437}a^{8}+\frac{2176003}{304311}a^{7}+\frac{379840}{101437}a^{6}-\frac{1210023}{101437}a^{5}+\frac{1239628}{304311}a^{4}+\frac{1170989}{304311}a^{3}-\frac{637445}{304311}a^{2}-\frac{56893}{304311}a-\frac{122326}{101437}$, $\frac{495562}{304311}a^{15}-\frac{1943254}{304311}a^{14}-\frac{25501}{43473}a^{13}+\frac{19941253}{304311}a^{12}-\frac{15352710}{101437}a^{11}+\frac{6310746}{101437}a^{10}+\frac{116936105}{304311}a^{9}-\frac{15966573}{14491}a^{8}+\frac{12366706}{7077}a^{7}-\frac{585168425}{304311}a^{6}+\frac{453033215}{304311}a^{5}-\frac{78968691}{101437}a^{4}+\frac{11983013}{43473}a^{3}-\frac{27372731}{304311}a^{2}+\frac{11160598}{304311}a-\frac{888346}{101437}$, $\frac{32099}{101437}a^{15}-\frac{627470}{304311}a^{14}+\frac{706691}{304311}a^{13}+\frac{4700053}{304311}a^{12}-\frac{18304708}{304311}a^{11}+\frac{5961512}{101437}a^{10}+\frac{27820187}{304311}a^{9}-\frac{120939191}{304311}a^{8}+\frac{218386039}{304311}a^{7}-\frac{265385539}{304311}a^{6}+\frac{225323986}{304311}a^{5}-\frac{126829393}{304311}a^{4}+\frac{14774939}{101437}a^{3}-\frac{13153082}{304311}a^{2}+\frac{887278}{43473}a-\frac{425965}{101437}$, $\frac{529640}{304311}a^{15}-\frac{12177}{2359}a^{14}-\frac{1648498}{304311}a^{13}+\frac{19467976}{304311}a^{12}-\frac{235832}{2359}a^{11}-\frac{6567622}{304311}a^{10}+\frac{111321424}{304311}a^{9}-\frac{247468252}{304311}a^{8}+\frac{49586846}{43473}a^{7}-\frac{341620807}{304311}a^{6}+\frac{78137827}{101437}a^{5}-\frac{113650015}{304311}a^{4}+\frac{43509155}{304311}a^{3}-\frac{15098812}{304311}a^{2}+\frac{1305163}{101437}a-\frac{91307}{43473}$, $\frac{14686}{14491}a^{15}-\frac{222704}{304311}a^{14}-\frac{2825881}{304311}a^{13}+\frac{8652257}{304311}a^{12}+\frac{7127747}{304311}a^{11}-\frac{36862106}{304311}a^{10}+\frac{46025138}{304311}a^{9}-\frac{760204}{304311}a^{8}-\frac{82556128}{304311}a^{7}+\frac{24217220}{43473}a^{6}-\frac{63617119}{101437}a^{5}+\frac{41522063}{101437}a^{4}-\frac{44972309}{304311}a^{3}+\frac{13051454}{304311}a^{2}-\frac{6140903}{304311}a+\frac{1244603}{304311}$, $\frac{690874}{304311}a^{15}-\frac{2060512}{304311}a^{14}-\frac{108838}{14491}a^{13}+\frac{8631375}{101437}a^{12}-\frac{39437590}{304311}a^{11}-\frac{4683724}{101437}a^{10}+\frac{153496262}{304311}a^{9}-\frac{15244707}{14491}a^{8}+\frac{422854370}{304311}a^{7}-\frac{383510114}{304311}a^{6}+\frac{74551206}{101437}a^{5}-\frac{74582444}{304311}a^{4}+\frac{714509}{14491}a^{3}-\frac{1878512}{101437}a^{2}+\frac{654744}{101437}a-\frac{247309}{304311}$ 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:  \( 3323.42469139 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 3323.42469139 \cdot 1}{24\cdot\sqrt{1082535236962615296}}\cr\approx \mathstrut & 0.323290189970 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 40*x^13 - 96*x^12 + 48*x^11 + 226*x^10 - 692*x^9 + 1143*x^8 - 1312*x^7 + 1078*x^6 - 624*x^5 + 258*x^4 - 88*x^3 + 30*x^2 - 8*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 + 40*x^13 - 96*x^12 + 48*x^11 + 226*x^10 - 692*x^9 + 1143*x^8 - 1312*x^7 + 1078*x^6 - 624*x^5 + 258*x^4 - 88*x^3 + 30*x^2 - 8*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 + 40*x^13 - 96*x^12 + 48*x^11 + 226*x^10 - 692*x^9 + 1143*x^8 - 1312*x^7 + 1078*x^6 - 624*x^5 + 258*x^4 - 88*x^3 + 30*x^2 - 8*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 + 40*x^13 - 96*x^12 + 48*x^11 + 226*x^10 - 692*x^9 + 1143*x^8 - 1312*x^7 + 1078*x^6 - 624*x^5 + 258*x^4 - 88*x^3 + 30*x^2 - 8*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

$C_2^2\times D_4$ (as 16T25):

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 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), 4.0.4032.2, 4.0.4032.1, 4.0.1008.1, 4.0.1008.2, \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{24})\), 8.0.260112384.8, 8.0.16257024.2, 8.0.260112384.7, 8.0.260112384.3, 8.0.260112384.2, 8.0.260112384.6

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 16 siblings: 16.0.2599167103947239325696.9, 16.8.2599167103947239325696.2, 16.0.2599167103947239325696.10, 16.0.2599167103947239325696.11, 16.0.32088482764780732416.2, 16.0.162447943996702457856.14, 16.0.162447943996702457856.16
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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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$$8$$2$$36$
\(3\) 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}$
\(7\) Copy content Toggle raw display 7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$