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Properties

Label 16.0.103670069459943424.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.037\times 10^{17}$
Root discriminant \(11.57\)
Ramified primes $2,17$
Class number $1$
Class group trivial
Galois group $C_2^2\wr C_2$ (as 16T46)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 12*x^13 + 24*x^12 + 80*x^11 + 38*x^10 - 152*x^9 - 287*x^8 - 120*x^7 + 250*x^6 + 472*x^5 + 404*x^4 + 208*x^3 + 66*x^2 + 12*x + 1)
 
gp: K = bnfinit(y^16 - 8*y^14 - 12*y^13 + 24*y^12 + 80*y^11 + 38*y^10 - 152*y^9 - 287*y^8 - 120*y^7 + 250*y^6 + 472*y^5 + 404*y^4 + 208*y^3 + 66*y^2 + 12*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 - 12*x^13 + 24*x^12 + 80*x^11 + 38*x^10 - 152*x^9 - 287*x^8 - 120*x^7 + 250*x^6 + 472*x^5 + 404*x^4 + 208*x^3 + 66*x^2 + 12*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 - 12*x^13 + 24*x^12 + 80*x^11 + 38*x^10 - 152*x^9 - 287*x^8 - 120*x^7 + 250*x^6 + 472*x^5 + 404*x^4 + 208*x^3 + 66*x^2 + 12*x + 1)
 

\( x^{16} - 8 x^{14} - 12 x^{13} + 24 x^{12} + 80 x^{11} + 38 x^{10} - 152 x^{9} - 287 x^{8} - 120 x^{7} + \cdots + 1 \) 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:   \(103670069459943424\) \(\medspace = 2^{32}\cdot 17^{6}\) 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:  \(11.57\)
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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}+\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{13}+\frac{1}{8}a^{11}+\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}-\frac{3}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{1784}a^{15}+\frac{81}{1784}a^{14}+\frac{43}{892}a^{13}+\frac{41}{1784}a^{12}+\frac{1}{8}a^{11}+\frac{10}{223}a^{10}-\frac{395}{1784}a^{9}+\frac{411}{1784}a^{8}+\frac{3}{8}a^{7}-\frac{283}{892}a^{6}-\frac{327}{1784}a^{5}-\frac{147}{1784}a^{4}-\frac{65}{892}a^{3}+\frac{605}{1784}a^{2}+\frac{457}{1784}a+\frac{117}{892}$ 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

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:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{17183}{1784} a^{15} - \frac{1987}{446} a^{14} - \frac{133437}{1784} a^{13} - \frac{144681}{1784} a^{12} + 267 a^{11} + \frac{1152309}{1784} a^{10} + \frac{131513}{1784} a^{9} - \frac{1325499}{892} a^{8} - \frac{8327}{4} a^{7} - \frac{403063}{1784} a^{6} + \frac{4426863}{1784} a^{5} + \frac{761213}{223} a^{4} + \frac{4220499}{1784} a^{3} + \frac{1706743}{1784} a^{2} + \frac{191623}{892} a + \frac{38265}{1784} \)  (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:   $\frac{381}{223}a^{15}-\frac{941}{446}a^{14}-\frac{10541}{892}a^{13}-\frac{1104}{223}a^{12}+\frac{209}{4}a^{11}+\frac{33085}{446}a^{10}-\frac{22017}{446}a^{9}-\frac{102713}{446}a^{8}-\frac{739}{4}a^{7}+\frac{53733}{446}a^{6}+\frac{79681}{223}a^{5}+\frac{141983}{446}a^{4}+\frac{134373}{892}a^{3}+\frac{6836}{223}a^{2}-\frac{2191}{892}a-\frac{1207}{446}$, $\frac{12245}{1784}a^{15}-\frac{1743}{446}a^{14}-\frac{93819}{1784}a^{13}-\frac{93365}{1784}a^{12}+\frac{387}{2}a^{11}+\frac{780461}{1784}a^{10}+\frac{25963}{1784}a^{9}-\frac{928891}{892}a^{8}-\frac{5495}{4}a^{7}-\frac{110215}{1784}a^{6}+\frac{3064093}{1784}a^{5}+\frac{502898}{223}a^{4}+\frac{2722085}{1784}a^{3}+\frac{1089743}{1784}a^{2}+\frac{124547}{892}a+\frac{26317}{1784}$, $\frac{8223}{1784}a^{15}-\frac{9181}{1784}a^{14}-\frac{29079}{892}a^{13}-\frac{30807}{1784}a^{12}+\frac{1123}{8}a^{11}+\frac{194005}{892}a^{10}-\frac{201905}{1784}a^{9}-\frac{1149031}{1784}a^{8}-\frac{4519}{8}a^{7}+\frac{64198}{223}a^{6}+\frac{1798731}{1784}a^{5}+\frac{1684867}{1784}a^{4}+\frac{424851}{892}a^{3}+\frac{239733}{1784}a^{2}+\frac{39163}{1784}a+\frac{815}{446}$, $\frac{9037}{1784}a^{15}-\frac{6133}{1784}a^{14}-\frac{16997}{446}a^{13}-\frac{62103}{1784}a^{12}+\frac{1155}{8}a^{11}+\frac{272503}{892}a^{10}-\frac{23023}{1784}a^{9}-\frac{1339419}{1784}a^{8}-\frac{7499}{8}a^{7}+\frac{4295}{446}a^{6}+\frac{2178361}{1784}a^{5}+\frac{2760935}{1784}a^{4}+\frac{460595}{446}a^{3}+\frac{736217}{1784}a^{2}+\frac{171659}{1784}a+\frac{2586}{223}$, $\frac{4981}{1784}a^{15}-\frac{3291}{1784}a^{14}-\frac{18629}{892}a^{13}-\frac{35281}{1784}a^{12}+\frac{625}{8}a^{11}+\frac{151295}{892}a^{10}+\frac{2041}{1784}a^{9}-\frac{727821}{1784}a^{8}-\frac{4253}{8}a^{7}-\frac{5808}{223}a^{6}+\frac{1182797}{1784}a^{5}+\frac{1581641}{1784}a^{4}+\frac{550841}{892}a^{3}+\frac{451235}{1784}a^{2}+\frac{98053}{1784}a+\frac{1023}{223}$, $\frac{9935}{1784}a^{15}-\frac{4273}{892}a^{14}-\frac{73047}{1784}a^{13}-\frac{55167}{1784}a^{12}+\frac{655}{4}a^{11}+\frac{545709}{1784}a^{10}-\frac{120391}{1784}a^{9}-\frac{180527}{223}a^{8}-885a^{7}+\frac{289189}{1784}a^{6}+\frac{2323579}{1784}a^{5}+\frac{1307217}{892}a^{4}+\frac{1575561}{1784}a^{3}+\frac{559217}{1784}a^{2}+\frac{28269}{446}a+\frac{11165}{1784}$, $\frac{711}{446}a^{15}-\frac{2225}{1784}a^{14}-\frac{21901}{1784}a^{13}-\frac{8375}{892}a^{12}+\frac{399}{8}a^{11}+\frac{165749}{1784}a^{10}-\frac{24483}{892}a^{9}-\frac{458793}{1784}a^{8}-\frac{2045}{8}a^{7}+\frac{172735}{1784}a^{6}+\frac{96159}{223}a^{5}+\frac{722131}{1784}a^{4}+\frac{314667}{1784}a^{3}+\frac{21607}{892}a^{2}-\frac{13539}{1784}a-\frac{4173}{1784}$ 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:  \( 312.978531942 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 312.978531942 \cdot 1}{8\cdot\sqrt{103670069459943424}}\cr\approx \mathstrut & 0.295145940507 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 12*x^13 + 24*x^12 + 80*x^11 + 38*x^10 - 152*x^9 - 287*x^8 - 120*x^7 + 250*x^6 + 472*x^5 + 404*x^4 + 208*x^3 + 66*x^2 + 12*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 - 8*x^14 - 12*x^13 + 24*x^12 + 80*x^11 + 38*x^10 - 152*x^9 - 287*x^8 - 120*x^7 + 250*x^6 + 472*x^5 + 404*x^4 + 208*x^3 + 66*x^2 + 12*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 - 8*x^14 - 12*x^13 + 24*x^12 + 80*x^11 + 38*x^10 - 152*x^9 - 287*x^8 - 120*x^7 + 250*x^6 + 472*x^5 + 404*x^4 + 208*x^3 + 66*x^2 + 12*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 - 8*x^14 - 12*x^13 + 24*x^12 + 80*x^11 + 38*x^10 - 152*x^9 - 287*x^8 - 120*x^7 + 250*x^6 + 472*x^5 + 404*x^4 + 208*x^3 + 66*x^2 + 12*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

$C_2^2\wr C_2$ (as 16T46):

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 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), 4.0.4352.1, 4.0.1088.1, 4.0.4352.2, 4.0.272.1, 4.0.1088.2, \(\Q(\zeta_{8})\), 4.4.4352.1, 8.0.18939904.2, 8.0.18939904.3, 8.0.18939904.4

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 8 siblings: 8.4.342102016.1, 8.4.5473632256.1, 8.4.85525504.1, 8.0.21381376.1, 8.4.1368408064.1, 8.0.1368408064.1, 8.0.342102016.1, 8.0.85525504.1
Degree 16 siblings: 16.8.29960650073923649536.1, 16.0.117033789351264256.2, 16.0.29960650073923649536.1, 16.0.1872540629620228096.1, 16.0.1872540629620228096.2, 16.0.29960650073923649536.5
Minimal sibling: 8.0.21381376.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(17\) Copy content Toggle raw display 17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{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}$