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Properties

Label 16.0.101...216.12
Degree $16$
Signature $[0, 8]$
Discriminant $1.014\times 10^{20}$
Root discriminant \(17.80\)
Ramified primes $2,7$
Class number $2$
Class group [2]
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 14*x^12 - 84*x^11 + 232*x^10 - 104*x^9 - 207*x^8 - 44*x^7 + 324*x^6 + 288*x^5 - 696*x^4 + 292*x^3 - 424*x^2 + 420*x + 1513)
 
gp: K = bnfinit(y^16 - 8*y^15 + 24*y^14 - 28*y^13 + 14*y^12 - 84*y^11 + 232*y^10 - 104*y^9 - 207*y^8 - 44*y^7 + 324*y^6 + 288*y^5 - 696*y^4 + 292*y^3 - 424*y^2 + 420*y + 1513, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 14*x^12 - 84*x^11 + 232*x^10 - 104*x^9 - 207*x^8 - 44*x^7 + 324*x^6 + 288*x^5 - 696*x^4 + 292*x^3 - 424*x^2 + 420*x + 1513);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 14*x^12 - 84*x^11 + 232*x^10 - 104*x^9 - 207*x^8 - 44*x^7 + 324*x^6 + 288*x^5 - 696*x^4 + 292*x^3 - 424*x^2 + 420*x + 1513)
 

\( x^{16} - 8 x^{15} + 24 x^{14} - 28 x^{13} + 14 x^{12} - 84 x^{11} + 232 x^{10} - 104 x^{9} - 207 x^{8} + \cdots + 1513 \) 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:   \(101415451701035401216\) \(\medspace = 2^{44}\cdot 7^{8}\) 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:  \(17.80\)
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:  $2^{11/4}7^{1/2}\approx 17.798422345016238$
Ramified primes:   \(2\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{35}a^{12}-\frac{6}{35}a^{11}-\frac{6}{35}a^{10}+\frac{3}{7}a^{9}+\frac{11}{35}a^{8}+\frac{2}{7}a^{7}-\frac{17}{35}a^{6}-\frac{16}{35}a^{5}-\frac{17}{35}a^{4}-\frac{2}{5}a^{3}+\frac{13}{35}a^{2}-\frac{9}{35}a+\frac{2}{35}$, $\frac{1}{35}a^{13}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}-\frac{4}{35}a^{9}+\frac{6}{35}a^{8}+\frac{8}{35}a^{7}-\frac{13}{35}a^{6}-\frac{8}{35}a^{5}-\frac{11}{35}a^{4}-\frac{1}{35}a^{3}-\frac{1}{35}a^{2}-\frac{17}{35}a+\frac{12}{35}$, $\frac{1}{77683795}a^{14}-\frac{1}{11097685}a^{13}+\frac{683392}{77683795}a^{12}-\frac{4100261}{77683795}a^{11}+\frac{24981564}{77683795}a^{10}-\frac{9638466}{77683795}a^{9}-\frac{3937972}{15536759}a^{8}+\frac{13803856}{77683795}a^{7}+\frac{1186120}{15536759}a^{6}-\frac{2245832}{4569635}a^{5}+\frac{8693808}{77683795}a^{4}+\frac{4046508}{15536759}a^{3}+\frac{36682612}{77683795}a^{2}+\frac{7656632}{15536759}a-\frac{2006723}{4569635}$, $\frac{1}{801152977835}a^{15}+\frac{5149}{801152977835}a^{14}-\frac{7095212489}{801152977835}a^{13}-\frac{7582655183}{801152977835}a^{12}+\frac{35269472918}{160230595567}a^{11}-\frac{34054343709}{801152977835}a^{10}+\frac{3343769173}{9425329151}a^{9}-\frac{19529535941}{114450425405}a^{8}+\frac{213581886519}{801152977835}a^{7}+\frac{317037830416}{801152977835}a^{6}-\frac{163580778}{160230595567}a^{5}-\frac{1965184874}{22890085081}a^{4}+\frac{353343319512}{801152977835}a^{3}+\frac{164661264004}{801152977835}a^{2}-\frac{246682274477}{801152977835}a+\frac{415087759}{47126645755}$ 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:   \( -\frac{209851}{77683795} a^{14} + \frac{209851}{11097685} a^{13} - \frac{797897}{15536759} a^{12} + \frac{4840469}{77683795} a^{11} - \frac{6282406}{77683795} a^{10} + \frac{22051206}{77683795} a^{9} - \frac{39830283}{77683795} a^{8} + \frac{2891667}{11097685} a^{7} - \frac{1141862}{11097685} a^{6} + \frac{3265966}{4569635} a^{5} - \frac{5444166}{11097685} a^{4} - \frac{42009333}{77683795} a^{3} + \frac{66341104}{77683795} a^{2} - \frac{32041273}{77683795} a + \frac{1503048}{913927} \)  (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:   $\frac{1076413623}{801152977835}a^{15}-\frac{797884128}{114450425405}a^{14}+\frac{1080069570}{160230595567}a^{13}+\frac{21655154583}{801152977835}a^{12}-\frac{5092530411}{114450425405}a^{11}-\frac{38406754798}{801152977835}a^{10}+\frac{6671883949}{801152977835}a^{9}+\frac{315846104083}{801152977835}a^{8}-\frac{301866883023}{801152977835}a^{7}-\frac{202518066731}{801152977835}a^{6}-\frac{38614511479}{801152977835}a^{5}+\frac{558359359954}{801152977835}a^{4}-\frac{203985801582}{801152977835}a^{3}-\frac{531603809216}{801152977835}a^{2}-\frac{81436293125}{160230595567}a-\frac{17389115869}{9425329151}$, $\frac{565187821}{801152977835}a^{15}-\frac{3451289378}{801152977835}a^{14}+\frac{7579222747}{801152977835}a^{13}-\frac{5557449004}{801152977835}a^{12}+\frac{1730800006}{160230595567}a^{11}-\frac{47299976022}{801152977835}a^{10}+\frac{80619524509}{801152977835}a^{9}-\frac{36773979657}{801152977835}a^{8}+\frac{31014121438}{801152977835}a^{7}-\frac{178881470921}{801152977835}a^{6}+\frac{252960726959}{801152977835}a^{5}-\frac{54403276239}{801152977835}a^{4}+\frac{5820216823}{801152977835}a^{3}-\frac{454107058523}{801152977835}a^{2}-\frac{340189395428}{801152977835}a-\frac{276426811}{1346475593}$, $\frac{4912738}{47126645755}a^{15}-\frac{9557337}{47126645755}a^{14}-\frac{7033748}{9425329151}a^{13}+\frac{68290111}{47126645755}a^{12}+\frac{95585688}{47126645755}a^{11}+\frac{90549561}{9425329151}a^{10}-\frac{2328590792}{47126645755}a^{9}+\frac{906876601}{47126645755}a^{8}+\frac{640775306}{9425329151}a^{7}+\frac{545198679}{9425329151}a^{6}-\frac{10578299024}{47126645755}a^{5}-\frac{3208453062}{47126645755}a^{4}+\frac{2953622786}{9425329151}a^{3}-\frac{17438826876}{47126645755}a^{2}-\frac{7643459946}{47126645755}a+\frac{36453450717}{47126645755}$, $\frac{149619726}{114450425405}a^{15}-\frac{5985319654}{801152977835}a^{14}+\frac{1602805111}{114450425405}a^{13}-\frac{3039161116}{801152977835}a^{12}+\frac{9475818851}{801152977835}a^{11}-\frac{78773780346}{801152977835}a^{10}+\frac{95299237188}{801152977835}a^{9}+\frac{30558669696}{801152977835}a^{8}+\frac{1800625047}{114450425405}a^{7}-\frac{29454992792}{114450425405}a^{6}+\frac{84372628766}{801152977835}a^{5}-\frac{12270720}{1346475593}a^{4}-\frac{238146494223}{801152977835}a^{3}+\frac{2608393493}{160230595567}a^{2}-\frac{1029994575742}{801152977835}a-\frac{229319988}{343990115}$, $\frac{55776741}{114450425405}a^{15}+\frac{318011142}{801152977835}a^{14}-\frac{2259114756}{160230595567}a^{13}+\frac{31842600774}{801152977835}a^{12}-\frac{16612230063}{801152977835}a^{11}+\frac{1521631422}{160230595567}a^{10}-\frac{194298918953}{801152977835}a^{9}+\frac{343757800359}{801152977835}a^{8}+\frac{9527735381}{160230595567}a^{7}-\frac{26163027046}{160230595567}a^{6}-\frac{756537400156}{801152977835}a^{5}+\frac{511266173742}{801152977835}a^{4}+\frac{21166396786}{22890085081}a^{3}-\frac{97421277702}{114450425405}a^{2}-\frac{875951949924}{801152977835}a-\frac{63757964441}{47126645755}$, $\frac{21537444}{801152977835}a^{15}+\frac{55942175}{160230595567}a^{14}-\frac{150670711}{47126645755}a^{13}+\frac{105682254}{9425329151}a^{12}-\frac{2781107941}{114450425405}a^{11}+\frac{31772242099}{801152977835}a^{10}-\frac{30919354847}{801152977835}a^{9}+\frac{13440970203}{801152977835}a^{8}-\frac{62201312254}{801152977835}a^{7}+\frac{35282417666}{160230595567}a^{6}-\frac{61374527461}{801152977835}a^{5}-\frac{188414064918}{801152977835}a^{4}+\frac{19517466576}{160230595567}a^{3}-\frac{24366462426}{801152977835}a^{2}-\frac{17937824179}{114450425405}a-\frac{3371541044}{9425329151}$, $\frac{1556820358}{801152977835}a^{15}-\frac{2335230537}{160230595567}a^{14}+\frac{32526648751}{801152977835}a^{13}-\frac{34334901159}{801152977835}a^{12}+\frac{666490837}{22890085081}a^{11}-\frac{140209867963}{801152977835}a^{10}+\frac{341999997461}{801152977835}a^{9}-\frac{219106930998}{801152977835}a^{8}-\frac{2341285133}{801152977835}a^{7}-\frac{364190624722}{801152977835}a^{6}+\frac{777615458072}{801152977835}a^{5}-\frac{524846539733}{801152977835}a^{4}+\frac{15940964}{6732377965}a^{3}+\frac{178396039636}{801152977835}a^{2}-\frac{289572596926}{160230595567}a+\frac{40801475551}{47126645755}$ 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:  \( 5814.60059283 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 5814.60059283 \cdot 2}{8\cdot\sqrt{101415451701035401216}}\cr\approx \mathstrut & 0.350628149063 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 24*x^14 - 28*x^13 + 14*x^12 - 84*x^11 + 232*x^10 - 104*x^9 - 207*x^8 - 44*x^7 + 324*x^6 + 288*x^5 - 696*x^4 + 292*x^3 - 424*x^2 + 420*x + 1513)
 
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^15 + 24*x^14 - 28*x^13 + 14*x^12 - 84*x^11 + 232*x^10 - 104*x^9 - 207*x^8 - 44*x^7 + 324*x^6 + 288*x^5 - 696*x^4 + 292*x^3 - 424*x^2 + 420*x + 1513, 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^15 + 24*x^14 - 28*x^13 + 14*x^12 - 84*x^11 + 232*x^10 - 104*x^9 - 207*x^8 - 44*x^7 + 324*x^6 + 288*x^5 - 696*x^4 + 292*x^3 - 424*x^2 + 420*x + 1513);
 
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^15 + 24*x^14 - 28*x^13 + 14*x^12 - 84*x^11 + 232*x^10 - 104*x^9 - 207*x^8 - 44*x^7 + 324*x^6 + 288*x^5 - 696*x^4 + 292*x^3 - 424*x^2 + 420*x + 1513);
 
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 16T39):

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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.0.25088.2 x2, 4.4.7168.1, 4.2.50176.1 x2, 4.0.7168.1, 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\), 8.0.3211264.1, 8.4.10070523904.2, 8.0.10070523904.8, 8.4.2517630976.1, 8.0.2517630976.1, 8.0.10070523904.15, 8.0.205520896.4

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: 8.4.51380224.1, 8.0.51380224.1, 8.4.205520896.1, 8.4.10070523904.2, 8.0.205520896.2, 8.0.10070523904.8, 8.4.2517630976.1, 8.0.2517630976.1
Degree 16 siblings: 16.0.6338465731314712576.3, 16.0.42238838692642816.1, 16.0.101415451701035401216.4, 16.8.101415451701035401216.1, 16.0.101415451701035401216.10, 16.0.6338465731314712576.7
Minimal sibling: 8.4.51380224.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ R ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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 2.16.44.2$x^{16} + 8 x^{15} + 20 x^{14} + 16 x^{13} + 16 x^{12} + 8 x^{11} + 72 x^{10} + 136 x^{9} + 136 x^{8} + 160 x^{7} + 136 x^{6} + 208 x^{5} + 240 x^{4} + 208 x^{3} + 160 x^{2} + 48 x + 36$$8$$2$$44$$D_4\times C_2$$[2, 3, 7/2]^{2}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$