Properties

Label 16.0.281792804290560000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.818\times 10^{17}$
Root discriminant \(12.32\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^16 + 2*y^14 - 4*y^12 - 12*y^11 + 36*y^10 - 52*y^9 + 83*y^8 - 88*y^7 + 60*y^6 - 56*y^5 + 44*y^4 - 16*y^3 + 6*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*x + 1)
 

\( x^{16} + 2 x^{14} - 4 x^{12} - 12 x^{11} + 36 x^{10} - 52 x^{9} + 83 x^{8} - 88 x^{7} + 60 x^{6} - 56 x^{5} + 44 x^{4} - 16 x^{3} + 6 x^{2} - 4 x + 1 \) 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:   \(281792804290560000\) \(\medspace = 2^{36}\cdot 3^{8}\cdot 5^{4}\) 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:  \(12.32\)
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:  $2^{9/4}3^{1/2}5^{1/2}\approx 18.42311740638763$
Ramified primes:   \(2\), \(3\), \(5\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{55}a^{14}-\frac{16}{55}a^{13}-\frac{9}{55}a^{12}+\frac{16}{55}a^{11}-\frac{2}{55}a^{10}-\frac{17}{55}a^{9}+\frac{17}{55}a^{8}-\frac{4}{11}a^{7}-\frac{1}{5}a^{6}-\frac{17}{55}a^{5}+\frac{24}{55}a^{4}-\frac{26}{55}a^{3}-\frac{8}{55}a^{2}+\frac{14}{55}a-\frac{7}{55}$, $\frac{1}{5383015}a^{15}+\frac{45429}{5383015}a^{14}+\frac{1218441}{5383015}a^{13}+\frac{2340626}{5383015}a^{12}-\frac{1235062}{5383015}a^{11}+\frac{1589068}{5383015}a^{10}-\frac{82243}{5383015}a^{9}-\frac{315873}{1076603}a^{8}+\frac{2408909}{5383015}a^{7}-\frac{899982}{5383015}a^{6}-\frac{214536}{489365}a^{5}-\frac{763176}{5383015}a^{4}+\frac{2068647}{5383015}a^{3}+\frac{1258899}{5383015}a^{2}-\frac{2347857}{5383015}a+\frac{416120}{1076603}$ 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:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{7335514}{1076603} a^{15} + \frac{21771183}{5383015} a^{14} + \frac{88719952}{5383015} a^{13} + \frac{54921153}{5383015} a^{12} - \frac{107077247}{5383015} a^{11} - \frac{496850306}{5383015} a^{10} + \frac{1021962669}{5383015} a^{9} - \frac{1332800764}{5383015} a^{8} + \frac{462145676}{1076603} a^{7} - \frac{1932399318}{5383015} a^{6} + \frac{107675024}{489365} a^{5} - \frac{1432973573}{5383015} a^{4} + \frac{817209282}{5383015} a^{3} - \frac{168754859}{5383015} a^{2} + \frac{144507767}{5383015} a - \frac{68578216}{5383015} \)  (order $24$) 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:   $\frac{28548856}{5383015}a^{15}+\frac{3078458}{1076603}a^{14}+\frac{1204568}{97873}a^{13}+\frac{37307912}{5383015}a^{12}-\frac{8266331}{489365}a^{11}-\frac{387358399}{5383015}a^{10}+\frac{163829922}{1076603}a^{9}-\frac{96060953}{489365}a^{8}+\frac{1808031804}{5383015}a^{7}-\frac{1544640188}{5383015}a^{6}+\frac{905840102}{5383015}a^{5}-\frac{1100844652}{5383015}a^{4}+\frac{656515941}{5383015}a^{3}-\frac{116181614}{5383015}a^{2}+\frac{100583072}{5383015}a-\frac{53763817}{5383015}$, $\frac{21771183}{5383015}a^{15}+\frac{15364812}{5383015}a^{14}+\frac{54921153}{5383015}a^{13}+\frac{3603003}{489365}a^{12}-\frac{56719466}{5383015}a^{11}-\frac{298429851}{5383015}a^{10}+\frac{574432876}{5383015}a^{9}-\frac{146701986}{1076603}a^{8}+\frac{1295226842}{5383015}a^{7}-\frac{1016228936}{5383015}a^{6}+\frac{620970347}{5383015}a^{5}-\frac{796603798}{5383015}a^{4}+\frac{418086261}{5383015}a^{3}-\frac{75557653}{5383015}a^{2}+\frac{78132064}{5383015}a-\frac{7335514}{1076603}$, $\frac{10508290}{1076603}a^{15}+\frac{7408497}{1076603}a^{14}+\frac{26524677}{1076603}a^{13}+\frac{19065077}{1076603}a^{12}-\frac{27508077}{1076603}a^{11}-\frac{144248466}{1076603}a^{10}+\frac{276957070}{1076603}a^{9}-\frac{354610539}{1076603}a^{8}+\frac{627314593}{1076603}a^{7}-\frac{491222859}{1076603}a^{6}+\frac{300378937}{1076603}a^{5}-\frac{35088703}{97873}a^{4}+\frac{202376257}{1076603}a^{3}-\frac{36565289}{1076603}a^{2}+\frac{40461981}{1076603}a-\frac{17757919}{1076603}$, $\frac{14903225}{1076603}a^{15}+\frac{7611913}{1076603}a^{14}+\frac{33467510}{1076603}a^{13}+\frac{17048809}{1076603}a^{12}-\frac{51301040}{1076603}a^{11}-\frac{205104761}{1076603}a^{10}+\frac{432754775}{1076603}a^{9}-\frac{550959267}{1076603}a^{8}+\frac{947769389}{1076603}a^{7}-\frac{817918154}{1076603}a^{6}+\frac{462016375}{1076603}a^{5}-\frac{53099979}{97873}a^{4}+\frac{349781551}{1076603}a^{3}-\frac{52038528}{1076603}a^{2}+\frac{57040933}{1076603}a-\frac{29287961}{1076603}$, $\frac{575386}{5383015}a^{15}+\frac{5945118}{5383015}a^{14}+\frac{7343147}{5383015}a^{13}+\frac{3704353}{1076603}a^{12}+\frac{16315862}{5383015}a^{11}-\frac{2492058}{1076603}a^{10}-\frac{58696971}{5383015}a^{9}+\frac{96925508}{5383015}a^{8}-\frac{131593696}{5383015}a^{7}+\frac{274920889}{5383015}a^{6}-\frac{163323474}{5383015}a^{5}+\frac{25218667}{1076603}a^{4}-\frac{14282947}{489365}a^{3}+\frac{41863927}{5383015}a^{2}-\frac{22471621}{5383015}a+\frac{15882322}{5383015}$, $\frac{44958898}{5383015}a^{15}+\frac{4626477}{1076603}a^{14}+\frac{1855954}{97873}a^{13}+\frac{51904186}{5383015}a^{12}-\frac{13908363}{489365}a^{11}-\frac{619850747}{5383015}a^{10}+\frac{259459137}{1076603}a^{9}-\frac{152023424}{489365}a^{8}+\frac{2888037137}{5383015}a^{7}-\frac{2501099769}{5383015}a^{6}+\frac{1455860826}{5383015}a^{5}-\frac{1830800516}{5383015}a^{4}+\frac{1082490303}{5383015}a^{3}-\frac{198687312}{5383015}a^{2}+\frac{201199071}{5383015}a-\frac{92103701}{5383015}$, $\frac{68210074}{5383015}a^{15}+\frac{41492923}{5383015}a^{14}+\frac{165086467}{5383015}a^{13}+\frac{103560326}{5383015}a^{12}-\frac{200235361}{5383015}a^{11}-\frac{930987672}{5383015}a^{10}+\frac{1883578234}{5383015}a^{9}-\frac{2446934516}{5383015}a^{8}+\frac{386753806}{489365}a^{7}-\frac{703954011}{1076603}a^{6}+\frac{2129837717}{5383015}a^{5}-\frac{2631223961}{5383015}a^{4}+\frac{1467644821}{5383015}a^{3}-\frac{56709729}{1076603}a^{2}+\frac{52403763}{1076603}a-\frac{10652134}{489365}$ 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:  \( 1291.96804794 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1291.96804794 \cdot 1}{24\cdot\sqrt{281792804290560000}}\cr\approx \mathstrut & 0.246328424291 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*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 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*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 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*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 + 2*x^14 - 4*x^12 - 12*x^11 + 36*x^10 - 52*x^9 + 83*x^8 - 88*x^7 + 60*x^6 - 56*x^5 + 44*x^4 - 16*x^3 + 6*x^2 - 4*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

$C_2^2\times D_4$ (as 16T25):

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 $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), 4.0.1280.1, 4.0.2880.1, 4.0.320.1, 4.0.11520.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\zeta_{24})\), 8.0.530841600.2, 8.0.530841600.4, 8.0.6553600.1, 8.0.530841600.3, 8.0.132710400.4, 8.0.8294400.2

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 siblings: 16.0.11007531417600000000.6, 16.0.11007531417600000000.9, 16.0.176120502681600000000.4, 16.8.176120502681600000000.2, 16.0.176120502681600000000.5, 16.0.176120502681600000000.7, 16.0.176120502681600000000.10
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 R R R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 Deg $16$$8$$2$$36$
\(3\) Copy content Toggle raw display 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$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$