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Properties

Label 16.0.176...000.10
Degree $16$
Signature $[0, 8]$
Discriminant $1.761\times 10^{20}$
Root discriminant \(18.42\)
Ramified primes $2,3,5$
Class number $4$
Class group [4]
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 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625)
 
gp: K = bnfinit(y^16 + 2*y^14 - 24*y^13 + 8*y^12 - 36*y^11 + 228*y^10 - 136*y^9 + 227*y^8 - 976*y^7 + 708*y^6 - 572*y^5 + 1736*y^4 - 1240*y^3 + 450*y^2 - 1000*y + 625, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 2*x^14 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 2*x^14 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625)
 

\( x^{16} + 2 x^{14} - 24 x^{13} + 8 x^{12} - 36 x^{11} + 228 x^{10} - 136 x^{9} + 227 x^{8} - 976 x^{7} + \cdots + 625 \) 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:   \(176120502681600000000\) \(\medspace = 2^{36}\cdot 3^{8}\cdot 5^{8}\) 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:  \(18.42\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{1975}a^{14}-\frac{19}{395}a^{13}+\frac{2}{1975}a^{12}+\frac{11}{1975}a^{11}+\frac{663}{1975}a^{10}-\frac{346}{1975}a^{9}+\frac{198}{1975}a^{8}+\frac{979}{1975}a^{7}+\frac{472}{1975}a^{6}+\frac{284}{1975}a^{5}-\frac{322}{1975}a^{4}+\frac{843}{1975}a^{3}-\frac{499}{1975}a^{2}+\frac{73}{395}a-\frac{7}{79}$, $\frac{1}{50733414875}a^{15}+\frac{1106978}{10146682975}a^{14}+\frac{887020327}{50733414875}a^{13}+\frac{6463669506}{50733414875}a^{12}-\frac{15464168827}{50733414875}a^{11}+\frac{19055373284}{50733414875}a^{10}-\frac{6123318087}{50733414875}a^{9}-\frac{14655265791}{50733414875}a^{8}+\frac{18297044412}{50733414875}a^{7}+\frac{14012458104}{50733414875}a^{6}-\frac{2557388657}{50733414875}a^{5}-\frac{8266153777}{50733414875}a^{4}-\frac{8463474369}{50733414875}a^{3}+\frac{306995476}{2029336595}a^{2}-\frac{2617413}{2029336595}a-\frac{29431240}{405867319}$ 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

$C_{4}$, which has order $4$

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:   \( -\frac{3764225328}{50733414875} a^{15} - \frac{1075426871}{10146682975} a^{14} - \frac{15201206606}{50733414875} a^{13} + \frac{69267193662}{50733414875} a^{12} + \frac{69142392246}{50733414875} a^{11} + \frac{235228747643}{50733414875} a^{10} - \frac{537597767104}{50733414875} a^{9} - \frac{260682316757}{50733414875} a^{8} - \frac{1247795732776}{50733414875} a^{7} + \frac{2021600797493}{50733414875} a^{6} + \frac{225487607856}{50733414875} a^{5} + \frac{2592569776226}{50733414875} a^{4} - \frac{3274583465198}{50733414875} a^{3} + \frac{18056730218}{10146682975} a^{2} - \frac{69009682524}{2029336595} a + \frac{13791860111}{405867319} \)  (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:   $\frac{1719037319}{50733414875}a^{15}+\frac{493234012}{10146682975}a^{14}+\frac{6996616863}{50733414875}a^{13}-\frac{30023307036}{50733414875}a^{12}-\frac{26330205938}{50733414875}a^{11}-\frac{93764824529}{50733414875}a^{10}+\frac{243909884247}{50733414875}a^{9}+\frac{79438281946}{50733414875}a^{8}+\frac{435806838678}{50733414875}a^{7}-\frac{989901342074}{50733414875}a^{6}-\frac{37351877883}{50733414875}a^{5}-\frac{814517851838}{50733414875}a^{4}+\frac{1679198739139}{50733414875}a^{3}-\frac{2227176739}{2029336595}a^{2}+\frac{3970244924}{405867319}a-\frac{7240576648}{405867319}$, $\frac{3763987511}{50733414875}a^{15}+\frac{182347731}{2029336595}a^{14}+\frac{12887736047}{50733414875}a^{13}-\frac{75213389839}{50733414875}a^{12}-\frac{60681189212}{50733414875}a^{11}-\frac{205111612171}{50733414875}a^{10}+\frac{622683472183}{50733414875}a^{9}+\frac{235947022929}{50733414875}a^{8}+\frac{1111920095422}{50733414875}a^{7}-\frac{2437255117586}{50733414875}a^{6}-\frac{213552350937}{50733414875}a^{5}-\frac{2344378469242}{50733414875}a^{4}+\frac{4038937145371}{50733414875}a^{3}-\frac{14841483958}{10146682975}a^{2}+\frac{63043390864}{2029336595}a-\frac{17212977105}{405867319}$, $\frac{3764225328}{50733414875}a^{15}+\frac{1075426871}{10146682975}a^{14}+\frac{15201206606}{50733414875}a^{13}-\frac{69267193662}{50733414875}a^{12}-\frac{69142392246}{50733414875}a^{11}-\frac{235228747643}{50733414875}a^{10}+\frac{537597767104}{50733414875}a^{9}+\frac{260682316757}{50733414875}a^{8}+\frac{1247795732776}{50733414875}a^{7}-\frac{2021600797493}{50733414875}a^{6}-\frac{225487607856}{50733414875}a^{5}-\frac{2592569776226}{50733414875}a^{4}+\frac{3274583465198}{50733414875}a^{3}-\frac{18056730218}{10146682975}a^{2}+\frac{69009682524}{2029336595}a-\frac{13385992792}{405867319}$, $\frac{11026604}{2029336595}a^{15}+\frac{67609673}{2029336595}a^{14}+\frac{191261777}{2029336595}a^{13}+\frac{37239458}{405867319}a^{12}-\frac{640528277}{2029336595}a^{11}-\frac{2262969466}{2029336595}a^{10}-\frac{3239775174}{2029336595}a^{9}+\frac{2095422796}{2029336595}a^{8}+\frac{9482759417}{2029336595}a^{7}+\frac{14459085023}{2029336595}a^{6}-\frac{3980780798}{2029336595}a^{5}-\frac{17129629353}{2029336595}a^{4}-\frac{4845588651}{405867319}a^{3}+\frac{686996080}{405867319}a^{2}+\frac{12584526004}{2029336595}a+\frac{2540792750}{405867319}$, $\frac{51377647}{50733414875}a^{15}+\frac{559519056}{10146682975}a^{14}+\frac{7660856094}{50733414875}a^{13}+\frac{19912877407}{50733414875}a^{12}-\frac{20851819444}{50733414875}a^{11}-\frac{73794619752}{50733414875}a^{10}-\frac{239998319289}{50733414875}a^{9}+\frac{37116789048}{50733414875}a^{8}+\frac{209330920564}{50733414875}a^{7}+\frac{945889358538}{50733414875}a^{6}-\frac{27374103854}{50733414875}a^{5}-\frac{203734550144}{50733414875}a^{4}-\frac{1404284765693}{50733414875}a^{3}+\frac{601816768}{2029336595}a^{2}+\frac{455411512}{405867319}a+\frac{5557604431}{405867319}$, $\frac{5001141372}{50733414875}a^{15}+\frac{1076470797}{10146682975}a^{14}+\frac{16293328819}{50733414875}a^{13}-\frac{101567935083}{50733414875}a^{12}-\frac{68102069014}{50733414875}a^{11}-\frac{262303584562}{50733414875}a^{10}+\frac{842879873831}{50733414875}a^{9}+\frac{216134046913}{50733414875}a^{8}+\frac{1432002201209}{50733414875}a^{7}-\frac{3251693998427}{50733414875}a^{6}+\frac{27706420166}{50733414875}a^{5}-\frac{3005858409854}{50733414875}a^{4}+\frac{5251083215722}{50733414875}a^{3}-\frac{69881334894}{10146682975}a^{2}+\frac{78144501243}{2029336595}a-\frac{22037384775}{405867319}$, $\frac{372842027}{10146682975}a^{15}+\frac{780865036}{10146682975}a^{14}+\frac{1582457034}{10146682975}a^{13}-\frac{6211047526}{10146682975}a^{12}-\frac{11737274338}{10146682975}a^{11}-\frac{21796523579}{10146682975}a^{10}+\frac{1790280411}{405867319}a^{9}+\frac{65461315956}{10146682975}a^{8}+\frac{94345015573}{10146682975}a^{7}-\frac{32927649702}{2029336595}a^{6}-\frac{30860092587}{2029336595}a^{5}-\frac{138469509986}{10146682975}a^{4}+\frac{55751437864}{2029336595}a^{3}+\frac{107594603306}{10146682975}a^{2}+\frac{16831891808}{2029336595}a-\frac{7167569881}{405867319}$ 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:  \( 7183.55685667 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 7183.55685667 \cdot 4}{12\cdot\sqrt{176120502681600000000}}\cr\approx \mathstrut & 0.438280620115 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 2*x^14 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625)
 
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 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625, 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 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625);
 
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 - 24*x^13 + 8*x^12 - 36*x^11 + 228*x^10 - 136*x^9 + 227*x^8 - 976*x^7 + 708*x^6 - 572*x^5 + 1736*x^4 - 1240*x^3 + 450*x^2 - 1000*x + 625);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-3}) \), 4.0.1280.1, 4.0.2880.1, 4.0.320.1, 4.0.11520.1, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(i, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), 8.0.3317760000.2, 8.0.13271040000.3, 8.0.13271040000.4, 8.0.163840000.2, 8.0.13271040000.2, 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: 16.0.281792804290560000.1, 16.0.11007531417600000000.6, 16.0.11007531417600000000.9, 16.0.176120502681600000000.4, 16.8.176120502681600000000.2, 16.0.176120502681600000000.5, 16.0.176120502681600000000.7
Minimal sibling: 16.0.281792804290560000.1

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 R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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 5.4.2.1$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}$
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}$