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Properties

Label 16.0.110...944.2
Degree $16$
Signature $[0, 8]$
Discriminant $1.103\times 10^{22}$
Root discriminant \(23.86\)
Ramified primes $2,17$
Class number $4$
Class group [2, 2]
Galois group $C_2^2 : C_8$ (as 16T24)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 18*x^14 + 129*x^12 + 419*x^10 + 678*x^8 + 673*x^6 + 415*x^4 + 47*x^2 + 1)
 
gp: K = bnfinit(y^16 + 18*y^14 + 129*y^12 + 419*y^10 + 678*y^8 + 673*y^6 + 415*y^4 + 47*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 18*x^14 + 129*x^12 + 419*x^10 + 678*x^8 + 673*x^6 + 415*x^4 + 47*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 18*x^14 + 129*x^12 + 419*x^10 + 678*x^8 + 673*x^6 + 415*x^4 + 47*x^2 + 1)
 

\( x^{16} + 18x^{14} + 129x^{12} + 419x^{10} + 678x^{8} + 673x^{6} + 415x^{4} + 47x^{2} + 1 \) 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:   \(11034809241396899282944\) \(\medspace = 2^{16}\cdot 17^{14}\) 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:  \(23.86\)
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:  $2\cdot 17^{7/8}\approx 23.86012979730628$
Ramified primes:   \(2\), \(17\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{13}a^{10}+\frac{5}{13}a^{8}-\frac{1}{13}a^{6}-\frac{5}{13}a^{4}+\frac{1}{13}a^{2}+\frac{5}{13}$, $\frac{1}{13}a^{11}+\frac{5}{13}a^{9}-\frac{1}{13}a^{7}-\frac{5}{13}a^{5}+\frac{1}{13}a^{3}+\frac{5}{13}a$, $\frac{1}{52}a^{12}+\frac{1}{52}a^{10}-\frac{1}{2}a^{9}-\frac{21}{52}a^{8}-\frac{1}{52}a^{6}+\frac{2}{13}a^{4}-\frac{1}{2}a^{3}+\frac{7}{26}a^{2}-\frac{1}{2}a-\frac{7}{52}$, $\frac{1}{52}a^{13}+\frac{1}{52}a^{11}-\frac{1}{26}a^{10}-\frac{21}{52}a^{9}+\frac{4}{13}a^{8}-\frac{1}{52}a^{7}-\frac{6}{13}a^{6}+\frac{2}{13}a^{5}+\frac{5}{26}a^{4}+\frac{7}{26}a^{3}-\frac{1}{26}a^{2}-\frac{7}{52}a+\frac{4}{13}$, $\frac{1}{310076}a^{14}-\frac{1}{310076}a^{12}-\frac{1}{26}a^{11}-\frac{5815}{310076}a^{10}+\frac{4}{13}a^{9}-\frac{109727}{310076}a^{8}-\frac{6}{13}a^{7}-\frac{51465}{155038}a^{6}+\frac{5}{26}a^{5}+\frac{3221}{155038}a^{4}-\frac{1}{26}a^{3}-\frac{26575}{310076}a^{2}+\frac{4}{13}a-\frac{51627}{155038}$, $\frac{1}{310076}a^{15}-\frac{1}{310076}a^{13}-\frac{5815}{310076}a^{11}-\frac{1}{26}a^{10}-\frac{109727}{310076}a^{9}-\frac{5}{26}a^{8}-\frac{51465}{155038}a^{7}-\frac{6}{13}a^{6}+\frac{3221}{155038}a^{5}+\frac{5}{26}a^{4}-\frac{26575}{310076}a^{3}+\frac{6}{13}a^{2}-\frac{51627}{155038}a-\frac{5}{26}$ 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:  $2$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:   \( -1 \)  (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:   $\frac{72015}{310076}a^{15}-\frac{7}{1157}a^{14}+\frac{323378}{77519}a^{13}-\frac{253}{2314}a^{12}+\frac{4619513}{155038}a^{11}-\frac{1805}{2314}a^{10}+\frac{7448979}{77519}a^{9}-\frac{2829}{1157}a^{8}+\frac{47513163}{310076}a^{7}-\frac{7633}{2314}a^{6}+\frac{23064499}{155038}a^{5}-\frac{1929}{1157}a^{4}+\frac{27456025}{310076}a^{3}-\frac{415}{2314}a^{2}+\frac{1890609}{310076}a+\frac{899}{1157}$, $\frac{2950}{77519}a^{15}+\frac{107397}{155038}a^{13}+\frac{783755}{155038}a^{11}+\frac{2643293}{155038}a^{9}+\frac{4594319}{155038}a^{7}+\frac{2474464}{77519}a^{5}+\frac{1668951}{77519}a^{3}+\frac{869967}{155038}a$, $\frac{48903}{77519}a^{15}+\frac{1756687}{155038}a^{13}+\frac{12549239}{155038}a^{11}+\frac{40504651}{155038}a^{9}+\frac{64818129}{155038}a^{7}+\frac{31760811}{77519}a^{5}+\frac{19221596}{77519}a^{3}+\frac{3290133}{155038}a$, $\frac{1063}{155038}a^{14}+\frac{8413}{77519}a^{12}+\frac{48847}{77519}a^{10}+\frac{90666}{77519}a^{8}-\frac{35281}{155038}a^{6}-\frac{141951}{77519}a^{4}-\frac{187347}{155038}a^{2}+\frac{25791}{155038}$, $\frac{48553}{155038}a^{15}+\frac{869749}{155038}a^{13}+\frac{6190023}{155038}a^{11}+\frac{19840627}{155038}a^{9}+\frac{15703658}{77519}a^{7}+\frac{15227114}{77519}a^{5}+\frac{17914669}{155038}a^{3}+\frac{390285}{77519}a$, $\frac{43353}{155038}a^{15}+\frac{779541}{155038}a^{13}+\frac{5575461}{155038}a^{11}+\frac{18020731}{155038}a^{9}+\frac{14408247}{77519}a^{7}+\frac{14059233}{77519}a^{5}+\frac{17190621}{155038}a^{3}+\frac{819798}{77519}a$, $\frac{48903}{77519}a^{15}+\frac{1756687}{155038}a^{13}+\frac{12549239}{155038}a^{11}+\frac{40504651}{155038}a^{9}+\frac{64818129}{155038}a^{7}+\frac{31760811}{77519}a^{5}+\frac{19221596}{77519}a^{3}+\frac{3290133}{155038}a-1$ 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:  \( 5872.0012145 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5872.0012145 \cdot 4}{2\cdot\sqrt{11034809241396899282944}}\cr\approx \mathstrut & 0.27156429221 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 18*x^14 + 129*x^12 + 419*x^10 + 678*x^8 + 673*x^6 + 415*x^4 + 47*x^2 + 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 + 18*x^14 + 129*x^12 + 419*x^10 + 678*x^8 + 673*x^6 + 415*x^4 + 47*x^2 + 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 + 18*x^14 + 129*x^12 + 419*x^10 + 678*x^8 + 673*x^6 + 415*x^4 + 47*x^2 + 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 + 18*x^14 + 129*x^12 + 419*x^10 + 678*x^8 + 673*x^6 + 415*x^4 + 47*x^2 + 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

$C_2^2:C_8$ (as 16T24):

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 $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, 4.2.1156.1, 4.2.19652.1, 8.4.6565418768.1, 8.0.105046700288.1, 8.4.386201104.1

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

Galois closure: deg 32
Degree 16 sibling: 16.8.43104723599206637824.1
Minimal sibling: 16.8.43104723599206637824.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.8.0.1}{8} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(17\) Copy content Toggle raw display 17.8.7.3$x^{8} + 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} + 17$$8$$1$$7$$C_8$$[\ ]_{8}$