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Properties

Label 16.0.277...776.2
Degree $16$
Signature $[0, 8]$
Discriminant $2.771\times 10^{20}$
Root discriminant \(18.95\)
Ramified primes $2,3,7$
Class number $2$
Class group [2]
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 + 4*x^14 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*x^3 + 4*x^2 + 4*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 4*y^14 - 24*y^13 + 104*y^12 - 196*y^11 + 312*y^10 - 236*y^9 + 31*y^8 + 236*y^7 + 312*y^6 + 196*y^5 + 104*y^4 + 24*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 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*x^3 + 4*x^2 + 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 4*x^14 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*x^3 + 4*x^2 + 4*x + 1)
 

\( x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + \cdots + 1 \) 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:   \(277129020662429515776\) \(\medspace = 2^{44}\cdot 3^{8}\cdot 7^{4}\) 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:  \(18.95\)
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:  $2^{11/4}3^{1/2}7^{1/2}\approx 30.827771796137323$
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 a CM field.
Reflex fields:  unavailable$^{128}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{153}a^{12}-\frac{2}{51}a^{11}+\frac{1}{17}a^{10}+\frac{2}{153}a^{9}+\frac{8}{51}a^{8}+\frac{8}{17}a^{7}-\frac{5}{153}a^{6}-\frac{8}{17}a^{5}+\frac{25}{51}a^{4}-\frac{53}{153}a^{3}+\frac{20}{51}a^{2}+\frac{19}{51}a+\frac{52}{153}$, $\frac{1}{153}a^{13}+\frac{8}{51}a^{11}+\frac{5}{153}a^{10}-\frac{5}{51}a^{9}+\frac{4}{51}a^{8}+\frac{70}{153}a^{7}-\frac{1}{3}a^{6}-\frac{62}{153}a^{4}+\frac{16}{51}a^{3}-\frac{14}{51}a^{2}-\frac{65}{153}a+\frac{19}{51}$, $\frac{1}{136629}a^{14}+\frac{307}{136629}a^{13}-\frac{23}{45543}a^{12}-\frac{637}{136629}a^{11}-\frac{1612}{136629}a^{10}+\frac{4663}{45543}a^{9}+\frac{10192}{136629}a^{8}-\frac{32228}{136629}a^{7}+\frac{3531}{15181}a^{6}-\frac{5657}{136629}a^{5}-\frac{57326}{136629}a^{4}+\frac{22708}{45543}a^{3}+\frac{12571}{136629}a^{2}-\frac{10409}{136629}a+\frac{3175}{15181}$, $\frac{1}{55881261}a^{15}-\frac{49}{18627087}a^{14}-\frac{93904}{55881261}a^{13}+\frac{6585}{6209029}a^{12}+\frac{994608}{6209029}a^{11}+\frac{7196869}{55881261}a^{10}+\frac{344244}{6209029}a^{9}+\frac{1711676}{18627087}a^{8}+\frac{871799}{55881261}a^{7}+\frac{22586885}{55881261}a^{6}+\frac{6056569}{18627087}a^{5}-\frac{5418949}{55881261}a^{4}+\frac{11108855}{55881261}a^{3}+\frac{5239012}{18627087}a^{2}-\frac{858244}{55881261}a+\frac{26085877}{55881261}$ 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

$C_{2}$, 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:   \( -\frac{2789963}{55881261} a^{15} + \frac{5053132}{18627087} a^{14} - \frac{28325140}{55881261} a^{13} + \frac{88475296}{55881261} a^{12} - \frac{43826381}{6209029} a^{11} + \frac{995136808}{55881261} a^{10} - \frac{1797348019}{55881261} a^{9} + \frac{39117262}{980373} a^{8} - \frac{1586677297}{55881261} a^{7} + \frac{19522328}{18627087} a^{6} - \frac{86751463}{18627087} a^{5} + \frac{561664136}{55881261} a^{4} + \frac{26146436}{6209029} a^{3} + \frac{48922299}{6209029} a^{2} + \frac{15404543}{55881261} a + \frac{10671152}{55881261} \)  (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:   $\frac{1537112}{6209029}a^{15}-\frac{64112846}{55881261}a^{14}+\frac{93912434}{55881261}a^{13}-\frac{381979504}{55881261}a^{12}+\frac{1668044366}{55881261}a^{11}-\frac{3715509806}{55881261}a^{10}+\frac{6426826240}{55881261}a^{9}-\frac{6809102612}{55881261}a^{8}+\frac{12013606}{173007}a^{7}+\frac{1631311526}{55881261}a^{6}+\frac{2993148016}{55881261}a^{5}+\frac{323683055}{55881261}a^{4}+\frac{801585146}{55881261}a^{3}-\frac{66863192}{55881261}a^{2}+\frac{152483858}{55881261}a+\frac{2170831}{1188963}$, $\frac{50603248}{55881261}a^{15}-\frac{74037104}{18627087}a^{14}+\frac{287166497}{55881261}a^{13}-\frac{1319048276}{55881261}a^{12}+\frac{640858493}{6209029}a^{11}-\frac{12120173939}{55881261}a^{10}+\frac{1195160716}{3287133}a^{9}-\frac{20095918}{57669}a^{8}+\frac{8522463881}{55881261}a^{7}+\frac{3066883612}{18627087}a^{6}+\frac{3989840524}{18627087}a^{5}+\frac{4982698592}{55881261}a^{4}+\frac{942532993}{18627087}a^{3}-\frac{17689212}{6209029}a^{2}+\frac{215971925}{55881261}a+\frac{101976317}{55881261}$, $\frac{2789963}{55881261}a^{15}-\frac{5053132}{18627087}a^{14}+\frac{28325140}{55881261}a^{13}-\frac{88475296}{55881261}a^{12}+\frac{43826381}{6209029}a^{11}-\frac{995136808}{55881261}a^{10}+\frac{1797348019}{55881261}a^{9}-\frac{39117262}{980373}a^{8}+\frac{1586677297}{55881261}a^{7}-\frac{19522328}{18627087}a^{6}+\frac{86751463}{18627087}a^{5}-\frac{561664136}{55881261}a^{4}-\frac{26146436}{6209029}a^{3}-\frac{48922299}{6209029}a^{2}-\frac{15404543}{55881261}a+\frac{45210109}{55881261}$, $a$, $\frac{313562}{396321}a^{15}-\frac{180747941}{55881261}a^{14}+\frac{21551616}{6209029}a^{13}-\frac{1082384000}{55881261}a^{12}+\frac{4695610937}{55881261}a^{11}-\frac{9394700}{57669}a^{10}+\frac{14704029845}{55881261}a^{9}-\frac{699632467}{3287133}a^{8}+\frac{291421410}{6209029}a^{7}+\frac{10212597988}{55881261}a^{6}+\frac{12606853561}{55881261}a^{5}+\frac{2679439948}{18627087}a^{4}+\frac{253090700}{3287133}a^{3}+\frac{991995856}{55881261}a^{2}+\frac{53891425}{18627087}a+\frac{164272420}{55881261}$, $\frac{766355}{980373}a^{15}-\frac{203152751}{55881261}a^{14}+\frac{301557775}{55881261}a^{13}-\frac{1219278989}{55881261}a^{12}+\frac{5297920148}{55881261}a^{11}-\frac{11858715445}{55881261}a^{10}+\frac{440169343}{1188963}a^{9}-\frac{22280527145}{55881261}a^{8}+\frac{13259434282}{55881261}a^{7}+\frac{4468828774}{55881261}a^{6}+\frac{9130105525}{55881261}a^{5}+\frac{1921917871}{55881261}a^{4}+\frac{2394603868}{55881261}a^{3}-\frac{35280448}{3287133}a^{2}+\frac{286078342}{55881261}a+\frac{59017117}{55881261}$, $\frac{20268908}{55881261}a^{15}-\frac{26882090}{18627087}a^{14}+\frac{75008374}{55881261}a^{13}-\frac{469399603}{55881261}a^{12}+\frac{696520969}{18627087}a^{11}-\frac{3835842742}{55881261}a^{10}+\frac{5839697860}{55881261}a^{9}-\frac{1344881344}{18627087}a^{8}-\frac{323701373}{55881261}a^{7}+\frac{1669579511}{18627087}a^{6}+\frac{41105616}{326791}a^{5}+\frac{2896512607}{55881261}a^{4}+\frac{83674637}{18627087}a^{3}-\frac{78146481}{6209029}a^{2}-\frac{380132219}{55881261}a-\frac{79333409}{55881261}$ 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:  \( 27537.240161576334 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 27537.240161576334 \cdot 2}{24\cdot\sqrt{277129020662429515776}}\cr\approx \mathstrut & 0.334839827503306 \end{aligned}\]

# 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 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*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 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*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 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*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 - 24*x^13 + 104*x^12 - 196*x^11 + 312*x^10 - 236*x^9 + 31*x^8 + 236*x^7 + 312*x^6 + 196*x^5 + 104*x^4 + 24*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

$C_2^2\times D_4$ (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), 4.4.64512.1, 4.4.7168.1, 4.0.7168.1, 4.0.64512.5, \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\zeta_{8})\), \(\Q(\zeta_{24})\), 8.8.16647192576.1, 8.0.16647192576.31, 8.0.4161798144.10, 8.0.4161798144.13, 8.0.16647192576.11, 8.0.205520896.4

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: deg 16, deg 16, deg 16, deg 16, deg 16, deg 16, deg 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 ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.16.44.2$x^{16} + 8 x^{15} + 20 x^{14} + 16 x^{13} + 16 x^{12} + 8 x^{11} + 72 x^{10} + 136 x^{9} + 136 x^{8} + 160 x^{7} + 136 x^{6} + 208 x^{5} + 240 x^{4} + 208 x^{3} + 160 x^{2} + 48 x + 36$$8$$2$$44$$D_4\times C_2$$[2, 3, 7/2]^{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}$
\(7\) 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}$