Properties

Label 16.0.944...296.1
Degree $16$
Signature $[0, 8]$
Discriminant $9.441\times 10^{18}$
Root discriminant \(15.34\)
Ramified primes $2,3,7$
Class number $2$
Class group [2]
Galois group $D_8:C_2$ (as 16T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 - 6*x^13 + 86*x^12 - 218*x^11 + 226*x^10 - 288*x^9 + 724*x^8 - 930*x^7 + 340*x^6 + 248*x^5 - 223*x^4 + 30*x^3 + 20*x^2 - 8*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 2*y^14 - 6*y^13 + 86*y^12 - 218*y^11 + 226*y^10 - 288*y^9 + 724*y^8 - 930*y^7 + 340*y^6 + 248*y^5 - 223*y^4 + 30*y^3 + 20*y^2 - 8*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 2*x^14 - 6*x^13 + 86*x^12 - 218*x^11 + 226*x^10 - 288*x^9 + 724*x^8 - 930*x^7 + 340*x^6 + 248*x^5 - 223*x^4 + 30*x^3 + 20*x^2 - 8*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 2*x^14 - 6*x^13 + 86*x^12 - 218*x^11 + 226*x^10 - 288*x^9 + 724*x^8 - 930*x^7 + 340*x^6 + 248*x^5 - 223*x^4 + 30*x^3 + 20*x^2 - 8*x + 1)
 

\( x^{16} - 4 x^{15} + 2 x^{14} - 6 x^{13} + 86 x^{12} - 218 x^{11} + 226 x^{10} - 288 x^{9} + 724 x^{8} - 930 x^{7} + 340 x^{6} + 248 x^{5} - 223 x^{4} + 30 x^{3} + 20 x^{2} - 8 x + 1 \) 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:   \(9440732714731831296\) \(\medspace = 2^{24}\cdot 3^{14}\cdot 7^{6}\) 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:  \(15.34\)
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:  $2^{3/2}3^{7/8}7^{1/2}\approx 19.5692919025805$
Ramified primes:   \(2\), \(3\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{10}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a+\frac{2}{5}$, $\frac{1}{25}a^{14}+\frac{2}{25}a^{13}+\frac{1}{25}a^{11}+\frac{2}{25}a^{10}-\frac{12}{25}a^{8}+\frac{1}{5}a^{7}-\frac{8}{25}a^{6}-\frac{3}{25}a^{5}-\frac{11}{25}a^{4}-\frac{1}{25}a^{3}+\frac{4}{25}a+\frac{4}{25}$, $\frac{1}{27649375}a^{15}-\frac{63461}{27649375}a^{14}+\frac{1295679}{27649375}a^{13}-\frac{1720809}{27649375}a^{12}-\frac{1064826}{27649375}a^{11}-\frac{339172}{2126875}a^{10}+\frac{1803703}{27649375}a^{9}+\frac{5301066}{27649375}a^{8}+\frac{3611562}{27649375}a^{7}-\frac{4281139}{27649375}a^{6}+\frac{7598613}{27649375}a^{5}+\frac{8855357}{27649375}a^{4}+\frac{5978253}{27649375}a^{3}-\frac{7045716}{27649375}a^{2}+\frac{4076607}{27649375}a-\frac{12757657}{27649375}$ 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

$C_{2}$, 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:   \( \frac{81910604}{27649375} a^{15} - \frac{290760919}{27649375} a^{14} + \frac{27288541}{27649375} a^{13} - \frac{458947636}{27649375} a^{12} + \frac{6832608871}{27649375} a^{11} - \frac{1133996663}{2126875} a^{10} + \frac{11404450362}{27649375} a^{9} - \frac{17403695936}{27649375} a^{8} + \frac{50465170823}{27649375} a^{7} - \frac{51895498531}{27649375} a^{6} + \frac{690056552}{27649375} a^{5} + \frac{24877512228}{27649375} a^{4} - \frac{8532347713}{27649375} a^{3} - \frac{1926095839}{27649375} a^{2} + \frac{1210380228}{27649375} a - \frac{175251103}{27649375} \)  (order $12$) 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:   $\frac{74129726}{27649375}a^{15}-\frac{259941161}{27649375}a^{14}+\frac{22174879}{27649375}a^{13}-\frac{435327209}{27649375}a^{12}+\frac{6143576224}{27649375}a^{11}-\frac{1011386522}{2126875}a^{10}+\frac{10396441628}{27649375}a^{9}-\frac{16134636759}{27649375}a^{8}+\frac{45210026637}{27649375}a^{7}-\frac{46521591364}{27649375}a^{6}+\frac{2216244713}{27649375}a^{5}+\frac{21239026282}{27649375}a^{4}-\frac{8514255972}{27649375}a^{3}-\frac{1198643566}{27649375}a^{2}+\frac{1268153032}{27649375}a-\frac{220965132}{27649375}$, $\frac{30432676}{27649375}a^{15}-\frac{93123311}{27649375}a^{14}-\frac{38900971}{27649375}a^{13}-\frac{176561384}{27649375}a^{12}+\frac{2446122449}{27649375}a^{11}-\frac{327242722}{2126875}a^{10}+\frac{1864681603}{27649375}a^{9}-\frac{4850388409}{27649375}a^{8}+\frac{15600905287}{27649375}a^{7}-\frac{10547805689}{27649375}a^{6}-\frac{7401049712}{27649375}a^{5}+\frac{8763719007}{27649375}a^{4}-\frac{400557422}{27649375}a^{3}-\frac{1118555141}{27649375}a^{2}+\frac{302021007}{27649375}a-\frac{18116582}{27649375}$, $\frac{87228112}{27649375}a^{15}-\frac{288938132}{27649375}a^{14}-\frac{35270077}{27649375}a^{13}-\frac{507460983}{27649375}a^{12}+\frac{7144112613}{27649375}a^{11}-\frac{1079612514}{2126875}a^{10}+\frac{9133588111}{27649375}a^{9}-\frac{16763973108}{27649375}a^{8}+\frac{49779951944}{27649375}a^{7}-\frac{44079205818}{27649375}a^{6}-\frac{8011458469}{27649375}a^{5}+\frac{24204991734}{27649375}a^{4}-\frac{4675374039}{27649375}a^{3}-\frac{2508270317}{27649375}a^{2}+\frac{837084359}{27649375}a-\frac{95022334}{27649375}$, $a-1$, $\frac{28607393}{27649375}a^{15}-\frac{99766323}{27649375}a^{14}+\frac{6034672}{27649375}a^{13}-\frac{171087687}{27649375}a^{12}+\frac{2380167982}{27649375}a^{11}-\frac{385623321}{2126875}a^{10}+\frac{3914222279}{27649375}a^{9}-\frac{6432352137}{27649375}a^{8}+\frac{17754582991}{27649375}a^{7}-\frac{17748159552}{27649375}a^{6}+\frac{1216415359}{27649375}a^{5}+\frac{6385929326}{27649375}a^{4}-\frac{2031535421}{27649375}a^{3}-\frac{306632513}{27649375}a^{2}+\frac{170046076}{27649375}a-\frac{30432676}{27649375}$, $\frac{127958729}{27649375}a^{15}-\frac{460909894}{27649375}a^{14}+\frac{72462341}{27649375}a^{13}-\frac{734764886}{27649375}a^{12}+\frac{10697392896}{27649375}a^{11}-\frac{1818000688}{2126875}a^{10}+\frac{19483848862}{27649375}a^{9}-\frac{28754076486}{27649375}a^{8}+\frac{80435689073}{27649375}a^{7}-\frac{86271731231}{27649375}a^{6}+\frac{8056461977}{27649375}a^{5}+\frac{37670925703}{27649375}a^{4}-\frac{16577194488}{27649375}a^{3}-\frac{1807856964}{27649375}a^{2}+\frac{2330939953}{27649375}a-\frac{419555753}{27649375}$, $\frac{6140834}{2126875}a^{15}-\frac{21682724}{2126875}a^{14}+\frac{1736911}{2126875}a^{13}-\frac{34537831}{2126875}a^{12}+\frac{510969241}{2126875}a^{11}-\frac{1096166699}{2126875}a^{10}+\frac{841587427}{2126875}a^{9}-\frac{1290817831}{2126875}a^{8}+\frac{3740795958}{2126875}a^{7}-\frac{3819596426}{2126875}a^{6}-\frac{85508}{2126875}a^{5}+\frac{1919799363}{2126875}a^{4}-\frac{707898748}{2126875}a^{3}-\frac{129774519}{2126875}a^{2}+\frac{110372488}{2126875}a-\frac{17625063}{2126875}$ 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:  \( 2306.51987946 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 2306.51987946 \cdot 2}{12\cdot\sqrt{9440732714731831296}}\cr\approx \mathstrut & 0.303907940522 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 2*x^14 - 6*x^13 + 86*x^12 - 218*x^11 + 226*x^10 - 288*x^9 + 724*x^8 - 930*x^7 + 340*x^6 + 248*x^5 - 223*x^4 + 30*x^3 + 20*x^2 - 8*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 - 4*x^15 + 2*x^14 - 6*x^13 + 86*x^12 - 218*x^11 + 226*x^10 - 288*x^9 + 724*x^8 - 930*x^7 + 340*x^6 + 248*x^5 - 223*x^4 + 30*x^3 + 20*x^2 - 8*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 - 4*x^15 + 2*x^14 - 6*x^13 + 86*x^12 - 218*x^11 + 226*x^10 - 288*x^9 + 724*x^8 - 930*x^7 + 340*x^6 + 248*x^5 - 223*x^4 + 30*x^3 + 20*x^2 - 8*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 - 4*x^15 + 2*x^14 - 6*x^13 + 86*x^12 - 218*x^11 + 226*x^10 - 288*x^9 + 724*x^8 - 930*x^7 + 340*x^6 + 248*x^5 - 223*x^4 + 30*x^3 + 20*x^2 - 8*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

$D_8:C_2$ (as 16T45):

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 11 conjugacy class representatives for $D_8:C_2$
Character table for $D_8:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), 4.0.3024.2, 4.0.189.1, \(\Q(\zeta_{12})\), 8.0.9144576.3

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: 8.2.1344252672.1, 8.2.1344252672.2
Degree 16 siblings: 16.0.462595903021859733504.1, 16.4.462595903021859733504.1, 16.0.1807015246179139584.1
Minimal sibling: 8.2.1344252672.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.2.0.1}{2} }^{8}$ R ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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$$4$$4$$24$
\(3\) Copy content Toggle raw display 3.16.14.1$x^{16} + 16 x^{15} + 128 x^{14} + 672 x^{13} + 2576 x^{12} + 7616 x^{11} + 17920 x^{10} + 34182 x^{9} + 53410 x^{8} + 68544 x^{7} + 71344 x^{6} + 57904 x^{5} + 34832 x^{4} + 16128 x^{3} + 7241 x^{2} + 2966 x + 634$$8$$2$$14$$QD_{16}$$[\ ]_{8}^{2}$
\(7\) Copy content Toggle raw display 7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
7.4.2.2$x^{4} - 42 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$