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Properties

Label 16.0.308...281.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.085\times 10^{19}$
Root discriminant \(16.52\)
Ramified primes $3,7,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_4\times C_2$ (as 16T9)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 - 19*x^13 + 33*x^12 + 47*x^11 + 108*x^10 - 175*x^9 - 197*x^8 - 350*x^7 + 432*x^6 + 376*x^5 + 528*x^4 - 608*x^3 - 256*x^2 - 128*x + 256)
 
gp: K = bnfinit(y^16 - y^15 - 4*y^14 - 19*y^13 + 33*y^12 + 47*y^11 + 108*y^10 - 175*y^9 - 197*y^8 - 350*y^7 + 432*y^6 + 376*y^5 + 528*y^4 - 608*y^3 - 256*y^2 - 128*y + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 4*x^14 - 19*x^13 + 33*x^12 + 47*x^11 + 108*x^10 - 175*x^9 - 197*x^8 - 350*x^7 + 432*x^6 + 376*x^5 + 528*x^4 - 608*x^3 - 256*x^2 - 128*x + 256);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 4*x^14 - 19*x^13 + 33*x^12 + 47*x^11 + 108*x^10 - 175*x^9 - 197*x^8 - 350*x^7 + 432*x^6 + 376*x^5 + 528*x^4 - 608*x^3 - 256*x^2 - 128*x + 256)
 

\( x^{16} - x^{15} - 4 x^{14} - 19 x^{13} + 33 x^{12} + 47 x^{11} + 108 x^{10} - 175 x^{9} - 197 x^{8} + \cdots + 256 \) 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:   \(30853268336830129281\) \(\medspace = 3^{8}\cdot 7^{8}\cdot 13^{8}\) 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:  \(16.52\)
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:  $3^{1/2}7^{1/2}13^{1/2}\approx 16.522711641858304$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{8}-\frac{1}{2}a^{6}-\frac{1}{6}a^{5}+\frac{1}{6}a^{4}-\frac{1}{2}a^{2}+\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{12}a^{10}-\frac{1}{12}a^{9}-\frac{1}{4}a^{7}+\frac{5}{12}a^{6}-\frac{5}{12}a^{5}-\frac{1}{4}a^{3}-\frac{5}{12}a^{2}+\frac{1}{6}a$, $\frac{1}{24}a^{11}-\frac{1}{24}a^{10}+\frac{1}{24}a^{8}-\frac{7}{24}a^{7}-\frac{5}{24}a^{6}+\frac{5}{24}a^{4}-\frac{5}{24}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{144}a^{12}+\frac{1}{144}a^{11}+\frac{1}{72}a^{10}+\frac{5}{144}a^{9}+\frac{1}{48}a^{8}+\frac{41}{144}a^{7}+\frac{5}{72}a^{6}+\frac{25}{144}a^{5}+\frac{23}{48}a^{4}+\frac{1}{9}a^{3}+\frac{17}{36}a^{2}-\frac{4}{9}a-\frac{2}{9}$, $\frac{1}{288}a^{13}-\frac{1}{288}a^{12}+\frac{1}{288}a^{10}-\frac{7}{288}a^{9}+\frac{35}{288}a^{8}+\frac{1}{4}a^{7}+\frac{5}{288}a^{6}+\frac{19}{288}a^{5}-\frac{61}{144}a^{4}-\frac{3}{8}a^{3}-\frac{7}{36}a^{2}+\frac{1}{3}a+\frac{2}{9}$, $\frac{1}{569664}a^{14}-\frac{85}{63296}a^{13}+\frac{59}{35604}a^{12}-\frac{1691}{569664}a^{11}+\frac{925}{63296}a^{10}-\frac{17381}{569664}a^{9}-\frac{61}{8901}a^{8}-\frac{252391}{569664}a^{7}+\frac{30231}{63296}a^{6}-\frac{104981}{284832}a^{5}-\frac{1061}{8901}a^{4}+\frac{5519}{71208}a^{3}+\frac{3439}{8901}a^{2}-\frac{541}{17802}a-\frac{1976}{8901}$, $\frac{1}{19368576}a^{15}+\frac{5}{19368576}a^{14}+\frac{8581}{9684288}a^{13}+\frac{40801}{19368576}a^{12}+\frac{146239}{19368576}a^{11}-\frac{660679}{19368576}a^{10}-\frac{797219}{9684288}a^{9}-\frac{560009}{6456192}a^{8}+\frac{6816689}{19368576}a^{7}+\frac{960155}{2421072}a^{6}-\frac{287497}{2421072}a^{5}-\frac{35099}{201756}a^{4}+\frac{271657}{605268}a^{3}+\frac{255575}{605268}a^{2}+\frac{112153}{302634}a-\frac{64504}{151317}$ 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:  $2$, $3$

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:   \( \frac{1507}{140352} a^{15} + \frac{497}{140352} a^{14} - \frac{995}{23392} a^{13} - \frac{11953}{46784} a^{12} + \frac{1265}{46784} a^{11} + \frac{88159}{140352} a^{10} + \frac{130057}{70176} a^{9} + \frac{62657}{140352} a^{8} - \frac{100613}{46784} a^{7} - \frac{102701}{17544} a^{6} - \frac{165493}{70176} a^{5} + \frac{48647}{17544} a^{4} + \frac{68665}{8772} a^{3} + \frac{3491}{1462} a^{2} - \frac{4801}{2193} a - \frac{5062}{2193} \)  (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:   $\frac{17975}{19368576}a^{15}-\frac{39971}{19368576}a^{14}-\frac{32873}{4842144}a^{13}-\frac{15775}{842112}a^{12}+\frac{1169443}{19368576}a^{11}+\frac{2383913}{19368576}a^{10}+\frac{731141}{4842144}a^{9}-\frac{2536591}{6456192}a^{8}-\frac{18834007}{19368576}a^{7}-\frac{10173685}{9684288}a^{6}+\frac{2151553}{4842144}a^{5}+\frac{1825757}{807024}a^{4}+\frac{154749}{67252}a^{3}+\frac{5881}{8772}a^{2}-\frac{155185}{100878}a-\frac{131140}{151317}$, $\frac{4587}{538016}a^{15}+\frac{4951}{1076032}a^{14}-\frac{14327}{421056}a^{13}-\frac{1025807}{4842144}a^{12}-\frac{129317}{9684288}a^{11}+\frac{5093683}{9684288}a^{10}+\frac{1707579}{1076032}a^{9}+\frac{2559761}{4842144}a^{8}-\frac{16774837}{9684288}a^{7}-\frac{48839401}{9684288}a^{6}-\frac{1349543}{538016}a^{5}+\frac{5316743}{2421072}a^{4}+\frac{372391}{52632}a^{3}+\frac{711629}{302634}a^{2}-\frac{579401}{302634}a-\frac{3104}{1173}$, $\frac{89507}{19368576}a^{15}+\frac{102197}{19368576}a^{14}-\frac{92399}{4842144}a^{13}-\frac{752731}{6456192}a^{12}-\frac{1102933}{19368576}a^{11}+\frac{5492429}{19368576}a^{10}+\frac{417529}{538016}a^{9}+\frac{11381819}{19368576}a^{8}-\frac{7232711}{19368576}a^{7}-\frac{17493397}{9684288}a^{6}-\frac{441139}{269008}a^{5}-\frac{896623}{2421072}a^{4}+\frac{1729789}{1210536}a^{3}+\frac{68636}{151317}a^{2}+\frac{38033}{151317}a-\frac{5170}{6579}$, $\frac{23485}{4842144}a^{15}+\frac{14575}{4842144}a^{14}-\frac{16189}{807024}a^{13}-\frac{545165}{4842144}a^{12}-\frac{21265}{1614048}a^{11}+\frac{467915}{1614048}a^{10}+\frac{578845}{807024}a^{9}+\frac{616885}{1614048}a^{8}-\frac{1322075}{1614048}a^{7}-\frac{339985}{201756}a^{6}-\frac{841465}{807024}a^{5}+\frac{325555}{201756}a^{4}+\frac{332455}{302634}a^{3}+\frac{26615}{151317}a^{2}-\frac{40910}{16813}a+\frac{143747}{151317}$, $\frac{4959}{538016}a^{15}-\frac{2041}{1210536}a^{14}-\frac{2785}{70176}a^{13}-\frac{99863}{538016}a^{12}+\frac{340763}{2421072}a^{11}+\frac{1247249}{2421072}a^{10}+\frac{1588957}{1614048}a^{9}-\frac{1785335}{4842144}a^{8}-\frac{1075955}{605268}a^{7}-\frac{10272643}{4842144}a^{6}+\frac{90617}{100878}a^{5}+\frac{4318439}{1210536}a^{4}+\frac{56807}{52632}a^{3}-\frac{102637}{67252}a^{2}-\frac{415465}{100878}a+\frac{400376}{151317}$, $\frac{547}{67252}a^{15}-\frac{35993}{2421072}a^{14}-\frac{12775}{403512}a^{13}-\frac{24539}{201756}a^{12}+\frac{490363}{1210536}a^{11}+\frac{81679}{302634}a^{10}+\frac{343129}{807024}a^{9}-\frac{632291}{302634}a^{8}-\frac{1473949}{1210536}a^{7}-\frac{1586305}{1210536}a^{6}+\frac{1959187}{403512}a^{5}+\frac{7074997}{2421072}a^{4}+\frac{1709113}{1210536}a^{3}-\frac{1086433}{201756}a^{2}-\frac{109363}{50439}a+\frac{378098}{151317}$, $\frac{47267}{9684288}a^{15}-\frac{202231}{9684288}a^{14}+\frac{4387}{807024}a^{13}-\frac{383989}{9684288}a^{12}+\frac{10629}{25024}a^{11}-\frac{1502201}{3228096}a^{10}+\frac{124067}{807024}a^{9}-\frac{6711667}{3228096}a^{8}+\frac{2826469}{1076032}a^{7}-\frac{1649687}{1614048}a^{6}+\frac{4615969}{807024}a^{5}-\frac{723325}{100878}a^{4}+\frac{370241}{151317}a^{3}-\frac{1931153}{302634}a^{2}+\frac{167744}{16813}a-\frac{606355}{151317}$ 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:  \( 4784.98672664 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 4784.98672664 \cdot 1}{6\cdot\sqrt{30853268336830129281}}\cr\approx \mathstrut & 0.348752914545 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 - 19*x^13 + 33*x^12 + 47*x^11 + 108*x^10 - 175*x^9 - 197*x^8 - 350*x^7 + 432*x^6 + 376*x^5 + 528*x^4 - 608*x^3 - 256*x^2 - 128*x + 256)
 
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 - x^15 - 4*x^14 - 19*x^13 + 33*x^12 + 47*x^11 + 108*x^10 - 175*x^9 - 197*x^8 - 350*x^7 + 432*x^6 + 376*x^5 + 528*x^4 - 608*x^3 - 256*x^2 - 128*x + 256, 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 - x^15 - 4*x^14 - 19*x^13 + 33*x^12 + 47*x^11 + 108*x^10 - 175*x^9 - 197*x^8 - 350*x^7 + 432*x^6 + 376*x^5 + 528*x^4 - 608*x^3 - 256*x^2 - 128*x + 256);
 
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 - x^15 - 4*x^14 - 19*x^13 + 33*x^12 + 47*x^11 + 108*x^10 - 175*x^9 - 197*x^8 - 350*x^7 + 432*x^6 + 376*x^5 + 528*x^4 - 608*x^3 - 256*x^2 - 128*x + 256);
 
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

$C_2\times D_4$ (as 16T9):

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 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{273}) \), \(\Q(\sqrt{21}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{13}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-7}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{-91})\), \(\Q(\sqrt{-7}, \sqrt{13})\), 4.0.5733.1 x2, 4.0.117.1 x2, 4.2.24843.1 x2, 4.2.507.1 x2, 8.0.5554571841.1, 8.0.32867289.1 x2, 8.4.5554571841.1 x2, 8.0.5554571841.2, 8.0.2313441.1, 8.0.5554571841.3 x2, 8.0.617174649.1 x2

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 8 siblings: 8.0.617174649.1, 8.0.32867289.1, 8.4.5554571841.1, 8.0.5554571841.3
Minimal sibling: 8.0.32867289.1

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 ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.1.0.1}{1} }^{16}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(3\) 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 7.4.2.1$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 13.4.2.1$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}$