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Properties

Label 16.0.299...536.7
Degree $16$
Signature $[0, 8]$
Discriminant $2.996\times 10^{19}$
Root discriminant \(16.49\)
Ramified primes $2,17$
Class number $2$
Class group [2]
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 - 7*x^12 + 40*x^8 - 112*x^4 + 256)
 
gp: K = bnfinit(y^16 - 7*y^12 + 40*y^8 - 112*y^4 + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 7*x^12 + 40*x^8 - 112*x^4 + 256);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 7*x^12 + 40*x^8 - 112*x^4 + 256)
 

\( x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 \) 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:   \(29960650073923649536\) \(\medspace = 2^{32}\cdot 17^{8}\) 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:  \(16.49\)
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:  $2^{2}17^{1/2}\approx 16.492422502470642$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{9}-\frac{3}{8}a^{5}-\frac{1}{2}a$, $\frac{1}{16}a^{10}-\frac{3}{16}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{11}-\frac{3}{32}a^{7}-\frac{1}{2}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{128}a^{12}-\frac{1}{16}a^{9}+\frac{5}{128}a^{8}-\frac{1}{8}a^{7}+\frac{7}{16}a^{5}-\frac{7}{32}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{128}a^{13}+\frac{5}{128}a^{9}+\frac{9}{32}a^{5}$, $\frac{1}{512}a^{14}-\frac{1}{256}a^{13}-\frac{1}{256}a^{12}-\frac{11}{512}a^{10}+\frac{11}{256}a^{9}+\frac{11}{256}a^{8}+\frac{21}{128}a^{6}+\frac{11}{64}a^{5}-\frac{21}{64}a^{4}+\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{512}a^{15}-\frac{1}{256}a^{13}+\frac{5}{512}a^{11}-\frac{5}{256}a^{9}-\frac{7}{128}a^{7}-\frac{1}{4}a^{6}+\frac{7}{64}a^{5}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a$ 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

$C_{2}$, 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:   \( -\frac{3}{128} a^{13} + \frac{17}{128} a^{9} - \frac{19}{32} a^{5} + a \)  (order $8$) 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:   $\frac{1}{512}a^{15}+\frac{3}{512}a^{14}-\frac{1}{128}a^{13}+\frac{1}{256}a^{12}-\frac{11}{512}a^{11}-\frac{1}{512}a^{10}+\frac{3}{128}a^{9}-\frac{11}{256}a^{8}+\frac{5}{128}a^{7}+\frac{7}{128}a^{6}-\frac{7}{32}a^{5}+\frac{21}{64}a^{4}-\frac{1}{8}a^{2}-\frac{3}{4}$, $\frac{1}{512}a^{15}-\frac{3}{512}a^{14}-\frac{1}{128}a^{13}-\frac{1}{256}a^{12}-\frac{11}{512}a^{11}+\frac{1}{512}a^{10}+\frac{3}{128}a^{9}+\frac{11}{256}a^{8}+\frac{5}{128}a^{7}-\frac{7}{128}a^{6}-\frac{7}{32}a^{5}-\frac{21}{64}a^{4}+\frac{1}{8}a^{2}+\frac{3}{4}$, $\frac{1}{256}a^{15}-\frac{3}{256}a^{11}+\frac{1}{16}a^{10}-\frac{1}{64}a^{7}-\frac{3}{16}a^{6}-\frac{1}{8}a^{3}+\frac{3}{4}a^{2}+1$, $\frac{5}{512}a^{15}-\frac{5}{256}a^{13}+\frac{3}{128}a^{12}-\frac{23}{512}a^{11}-\frac{1}{16}a^{10}+\frac{23}{256}a^{9}-\frac{1}{128}a^{8}+\frac{17}{128}a^{7}+\frac{3}{16}a^{6}-\frac{17}{64}a^{5}+\frac{7}{32}a^{4}-\frac{1}{8}a^{3}-\frac{3}{4}a^{2}+\frac{1}{4}a+\frac{1}{2}$, $\frac{5}{512}a^{15}-\frac{5}{256}a^{13}-\frac{3}{128}a^{12}-\frac{23}{512}a^{11}+\frac{1}{16}a^{10}+\frac{23}{256}a^{9}+\frac{1}{128}a^{8}+\frac{17}{128}a^{7}-\frac{3}{16}a^{6}-\frac{17}{64}a^{5}-\frac{7}{32}a^{4}-\frac{1}{8}a^{3}+\frac{3}{4}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{5}{256}a^{15}+\frac{1}{128}a^{14}+\frac{1}{32}a^{12}-\frac{31}{256}a^{11}-\frac{11}{128}a^{10}-\frac{7}{32}a^{8}+\frac{23}{64}a^{7}+\frac{13}{32}a^{6}+\frac{3}{4}a^{4}-\frac{1}{8}a^{3}-\frac{3}{4}a^{2}-1$, $\frac{1}{256}a^{15}-\frac{1}{128}a^{14}+\frac{1}{128}a^{13}-\frac{11}{256}a^{11}+\frac{3}{128}a^{10}+\frac{5}{128}a^{9}+\frac{5}{64}a^{7}+\frac{1}{32}a^{6}-\frac{7}{32}a^{5}-\frac{3}{4}a^{2}+\frac{3}{2}a$ 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:  \( 22598.9729953 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 22598.9729953 \cdot 2}{8\cdot\sqrt{29960650073923649536}}\cr\approx \mathstrut & 2.50721768558 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^12 + 40*x^8 - 112*x^4 + 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 - 7*x^12 + 40*x^8 - 112*x^4 + 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 - 7*x^12 + 40*x^8 - 112*x^4 + 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 - 7*x^12 + 40*x^8 - 112*x^4 + 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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-17}) \), \(\Q(i, \sqrt{34})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{-2}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{-2}, \sqrt{17})\), 4.4.4352.1 x2, 4.0.1088.2 x2, 4.4.9248.1 x2, 4.0.2312.1 x2, 8.0.5473632256.1, 8.0.18939904.2 x2, 8.0.342102016.5 x2, 8.8.5473632256.1, 8.0.342102016.2, 8.0.5473632256.3 x2, 8.0.1368408064.5 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: 8.0.342102016.5, 8.0.5473632256.3, 8.0.18939904.2, 8.0.1368408064.5
Minimal sibling: 8.0.18939904.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.2.0.1}{2} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.8.2$x^{4} + 2 x^{2} + 4 x + 2$$4$$1$$8$$C_2^2$$[2, 3]$
\(17\) Copy content Toggle raw display 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$