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Properties

Label 16.4.524...125.2
Degree $16$
Signature $[4, 6]$
Discriminant $5.240\times 10^{19}$
Root discriminant \(17.08\)
Ramified primes $3,5,179,1021$
Class number $1$
Class group trivial
Galois group $C_4^4.C_2\wr C_4$ (as 16T1771)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 9*x^14 + 3*x^13 - 52*x^12 + 114*x^11 - 91*x^10 - 89*x^9 + 299*x^8 - 291*x^7 + 28*x^6 + 215*x^5 - 223*x^4 + 97*x^3 - 16*x^2 + x + 1)
 
gp: K = bnfinit(y^16 - 5*y^15 + 9*y^14 + 3*y^13 - 52*y^12 + 114*y^11 - 91*y^10 - 89*y^9 + 299*y^8 - 291*y^7 + 28*y^6 + 215*y^5 - 223*y^4 + 97*y^3 - 16*y^2 + y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 5*x^15 + 9*x^14 + 3*x^13 - 52*x^12 + 114*x^11 - 91*x^10 - 89*x^9 + 299*x^8 - 291*x^7 + 28*x^6 + 215*x^5 - 223*x^4 + 97*x^3 - 16*x^2 + x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 5*x^15 + 9*x^14 + 3*x^13 - 52*x^12 + 114*x^11 - 91*x^10 - 89*x^9 + 299*x^8 - 291*x^7 + 28*x^6 + 215*x^5 - 223*x^4 + 97*x^3 - 16*x^2 + x + 1)
 

\( x^{16} - 5 x^{15} + 9 x^{14} + 3 x^{13} - 52 x^{12} + 114 x^{11} - 91 x^{10} - 89 x^{9} + 299 x^{8} + \cdots + 1 \) 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:  $[4, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(52401279790283203125\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 179^{2}\cdot 1021\) 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:  \(17.08\)
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:  $3^{1/2}5^{3/4}179^{1/2}1021^{1/2}\approx 2475.868173489506$
Ramified primes:   \(3\), \(5\), \(179\), \(1021\) 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:  \(\Q(\sqrt{1021}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{386364509}a^{15}-\frac{39474325}{386364509}a^{14}+\frac{162914685}{386364509}a^{13}+\frac{48256188}{386364509}a^{12}-\frac{18398309}{386364509}a^{11}+\frac{164786933}{386364509}a^{10}+\frac{66688799}{386364509}a^{9}-\frac{23705760}{386364509}a^{8}+\frac{97117643}{386364509}a^{7}-\frac{145168194}{386364509}a^{6}-\frac{85206071}{386364509}a^{5}-\frac{88584030}{386364509}a^{4}+\frac{40217823}{386364509}a^{3}+\frac{166311228}{386364509}a^{2}+\frac{34893589}{386364509}a-\frac{18978718}{386364509}$ 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

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:  $9$
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:   $\frac{115825}{922111}a^{15}-\frac{578604}{922111}a^{14}+\frac{760733}{922111}a^{13}+\frac{1347143}{922111}a^{12}-\frac{6827478}{922111}a^{11}+\frac{10262573}{922111}a^{10}+\frac{450584}{922111}a^{9}-\frac{23496180}{922111}a^{8}+\frac{30997005}{922111}a^{7}-\frac{3058201}{922111}a^{6}-\frac{31037750}{922111}a^{5}+\frac{28495590}{922111}a^{4}-\frac{633947}{922111}a^{3}-\frac{11460220}{922111}a^{2}+\frac{3824917}{922111}a-\frac{664782}{922111}$, $\frac{35168338}{386364509}a^{15}-\frac{86633950}{386364509}a^{14}-\frac{5897388}{386364509}a^{13}+\frac{438066931}{386364509}a^{12}-\frac{1033608322}{386364509}a^{11}+\frac{377691674}{386364509}a^{10}+\frac{1972372740}{386364509}a^{9}-\frac{3729090387}{386364509}a^{8}+\frac{744311262}{386364509}a^{7}+\frac{4940095795}{386364509}a^{6}-\frac{6659351685}{386364509}a^{5}+\frac{905116619}{386364509}a^{4}+\frac{4707463298}{386364509}a^{3}-\frac{4639805456}{386364509}a^{2}+\frac{1374005313}{386364509}a+\frac{515334451}{386364509}$, $\frac{69037200}{386364509}a^{15}-\frac{305716094}{386364509}a^{14}+\frac{447697115}{386364509}a^{13}+\frac{464151583}{386364509}a^{12}-\frac{3301141007}{386364509}a^{11}+\frac{5847220472}{386364509}a^{10}-\frac{2716945324}{386364509}a^{9}-\frac{7620666093}{386364509}a^{8}+\frac{15068345031}{386364509}a^{7}-\frac{8939122257}{386364509}a^{6}-\frac{5142625578}{386364509}a^{5}+\frac{10848502783}{386364509}a^{4}-\frac{5767375609}{386364509}a^{3}+\frac{554668543}{386364509}a^{2}+\frac{14411430}{386364509}a+\frac{227153164}{386364509}$, $\frac{13362337}{386364509}a^{15}-\frac{155801126}{386364509}a^{14}+\frac{324644515}{386364509}a^{13}+\frac{126385986}{386364509}a^{12}-\frac{1827104960}{386364509}a^{11}+\frac{3922326413}{386364509}a^{10}-\frac{1783578044}{386364509}a^{9}-\frac{6115809033}{386364509}a^{8}+\frac{12243433833}{386364509}a^{7}-\frac{6221482014}{386364509}a^{6}-\frac{6345465565}{386364509}a^{5}+\frac{11034535591}{386364509}a^{4}-\frac{4973087091}{386364509}a^{3}-\frac{162026724}{386364509}a^{2}+\frac{228634910}{386364509}a-\frac{407513600}{386364509}$, $\frac{73670137}{386364509}a^{15}-\frac{203504050}{386364509}a^{14}+\frac{33139208}{386364509}a^{13}+\frac{838069943}{386364509}a^{12}-\frac{2156033469}{386364509}a^{11}+\frac{1306458292}{386364509}a^{10}+\frac{2816889417}{386364509}a^{9}-\frac{5778861346}{386364509}a^{8}+\frac{1450967775}{386364509}a^{7}+\frac{5778181350}{386364509}a^{6}-\frac{5473637679}{386364509}a^{5}-\frac{1636607635}{386364509}a^{4}+\frac{4702049981}{386364509}a^{3}-\frac{1289333161}{386364509}a^{2}-\frac{119351894}{386364509}a+\frac{311818582}{386364509}$, $\frac{26315994}{386364509}a^{15}-\frac{208170038}{386364509}a^{14}+\frac{553472146}{386364509}a^{13}-\frac{268871490}{386364509}a^{12}-\frac{2127539413}{386364509}a^{11}+\frac{6441157595}{386364509}a^{10}-\frac{7270347129}{386364509}a^{9}-\frac{2206207207}{386364509}a^{8}+\frac{17174516798}{386364509}a^{7}-\frac{19875816256}{386364509}a^{6}+\frac{3679371885}{386364509}a^{5}+\frac{12963210066}{386364509}a^{4}-\frac{13697145543}{386364509}a^{3}+\frac{5097720391}{386364509}a^{2}-\frac{686113546}{386364509}a-\frac{87344117}{386364509}$, $\frac{144225693}{386364509}a^{15}-\frac{633877692}{386364509}a^{14}+\frac{820697113}{386364509}a^{13}+\frac{1209321694}{386364509}a^{12}-\frac{6872398289}{386364509}a^{11}+\frac{11226111710}{386364509}a^{10}-\frac{3290667237}{386364509}a^{9}-\frac{17328444649}{386364509}a^{8}+\frac{29705885098}{386364509}a^{7}-\frac{15434234849}{386364509}a^{6}-\frac{10768657410}{386364509}a^{5}+\frac{20694977836}{386364509}a^{4}-\frac{12207684851}{386364509}a^{3}+\frac{4209547456}{386364509}a^{2}-\frac{1379839950}{386364509}a-\frac{73565624}{386364509}$, $\frac{9664897}{386364509}a^{15}-\frac{37221984}{386364509}a^{14}+\frac{48597691}{386364509}a^{13}+\frac{55976993}{386364509}a^{12}-\frac{450753085}{386364509}a^{11}+\frac{909990659}{386364509}a^{10}-\frac{627815277}{386364509}a^{9}-\frac{1120327756}{386364509}a^{8}+\frac{3873780806}{386364509}a^{7}-\frac{4701096724}{386364509}a^{6}+\frac{1391792692}{386364509}a^{5}+\frac{3885185023}{386364509}a^{4}-\frac{5920158836}{386364509}a^{3}+\frac{3303622158}{386364509}a^{2}-\frac{340773934}{386364509}a-\frac{31285278}{386364509}$, $\frac{159211345}{386364509}a^{15}-\frac{727019525}{386364509}a^{14}+\frac{1127186011}{386364509}a^{13}+\frac{925331150}{386364509}a^{12}-\frac{7814838357}{386364509}a^{11}+\frac{14848952323}{386364509}a^{10}-\frac{8641011923}{386364509}a^{9}-\frac{16886755029}{386364509}a^{8}+\frac{39983526062}{386364509}a^{7}-\frac{31262156364}{386364509}a^{6}-\frac{4481204597}{386364509}a^{5}+\frac{29087813597}{386364509}a^{4}-\frac{24655627152}{386364509}a^{3}+\frac{9676124856}{386364509}a^{2}-\frac{1992712977}{386364509}a+\frac{173622775}{386364509}$ 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:  \( 3051.12459213 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 3051.12459213 \cdot 1}{2\cdot\sqrt{52401279790283203125}}\cr\approx \mathstrut & 0.207471378515 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 9*x^14 + 3*x^13 - 52*x^12 + 114*x^11 - 91*x^10 - 89*x^9 + 299*x^8 - 291*x^7 + 28*x^6 + 215*x^5 - 223*x^4 + 97*x^3 - 16*x^2 + 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 - 5*x^15 + 9*x^14 + 3*x^13 - 52*x^12 + 114*x^11 - 91*x^10 - 89*x^9 + 299*x^8 - 291*x^7 + 28*x^6 + 215*x^5 - 223*x^4 + 97*x^3 - 16*x^2 + 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 - 5*x^15 + 9*x^14 + 3*x^13 - 52*x^12 + 114*x^11 - 91*x^10 - 89*x^9 + 299*x^8 - 291*x^7 + 28*x^6 + 215*x^5 - 223*x^4 + 97*x^3 - 16*x^2 + 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 - 5*x^15 + 9*x^14 + 3*x^13 - 52*x^12 + 114*x^11 - 91*x^10 - 89*x^9 + 299*x^8 - 291*x^7 + 28*x^6 + 215*x^5 - 223*x^4 + 97*x^3 - 16*x^2 + 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

$C_4^4.C_2\wr C_4$ (as 16T1771):

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 16384
The 190 conjugacy class representatives for $C_4^4.C_2\wr C_4$
Character table for $C_4^4.C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{15})^+\), 8.6.226546875.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
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 $16$ R R $16$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{7}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

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 3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(179\) Copy content Toggle raw display $\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
$\Q_{179}$$x + 177$$1$$1$$0$Trivial$[\ ]$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} + 172 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
179.4.0.1$x^{4} + x^{2} + 109 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
179.4.2.1$x^{4} + 344 x^{3} + 29946 x^{2} + 62264 x + 5326865$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(1021\) Copy content Toggle raw display $\Q_{1021}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1021}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$