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Properties

Label 16.12.673...248.2
Degree $16$
Signature $[12, 2]$
Discriminant $6.733\times 10^{20}$
Root discriminant \(20.03\)
Ramified primes $2,337$
Class number $1$
Class group trivial
Galois group $C_2^4.C_2\wr C_4$ (as 16T1251)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 4*x^13 + 48*x^12 + 32*x^11 - 78*x^10 - 84*x^9 + 78*x^8 + 84*x^7 - 88*x^6 + 12*x^5 + 37*x^4 - 36*x^3 + 6*x^2 + 4*x - 1)
 
gp: K = bnfinit(y^16 - 12*y^14 - 4*y^13 + 48*y^12 + 32*y^11 - 78*y^10 - 84*y^9 + 78*y^8 + 84*y^7 - 88*y^6 + 12*y^5 + 37*y^4 - 36*y^3 + 6*y^2 + 4*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 12*x^14 - 4*x^13 + 48*x^12 + 32*x^11 - 78*x^10 - 84*x^9 + 78*x^8 + 84*x^7 - 88*x^6 + 12*x^5 + 37*x^4 - 36*x^3 + 6*x^2 + 4*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 12*x^14 - 4*x^13 + 48*x^12 + 32*x^11 - 78*x^10 - 84*x^9 + 78*x^8 + 84*x^7 - 88*x^6 + 12*x^5 + 37*x^4 - 36*x^3 + 6*x^2 + 4*x - 1)
 

\( x^{16} - 12 x^{14} - 4 x^{13} + 48 x^{12} + 32 x^{11} - 78 x^{10} - 84 x^{9} + 78 x^{8} + 84 x^{7} + \cdots - 1 \) 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:  $[12, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(673301391207980597248\) \(\medspace = 2^{44}\cdot 337^{3}\) 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:  \(20.03\)
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:  $2^{11/4}337^{1/2}\approx 123.49444949726063$
Ramified primes:   \(2\), \(337\) 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:  \(\Q(\sqrt{337}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{9463247}a^{15}-\frac{4126723}{9463247}a^{14}+\frac{1072198}{9463247}a^{13}+\frac{9903}{9463247}a^{12}-\frac{4637275}{9463247}a^{11}+\frac{978011}{9463247}a^{10}-\frac{275001}{9463247}a^{9}+\frac{1444905}{9463247}a^{8}+\frac{2995734}{9463247}a^{7}+\frac{3826521}{9463247}a^{6}-\frac{3165516}{9463247}a^{5}-\frac{423425}{9463247}a^{4}-\frac{2482497}{9463247}a^{3}-\frac{2521260}{9463247}a^{2}+\frac{4378390}{9463247}a+\frac{1899062}{9463247}$ 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:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $a$, $\frac{4966494}{9463247}a^{15}+\frac{3921486}{9463247}a^{14}-\frac{55565040}{9463247}a^{13}-\frac{63547306}{9463247}a^{12}+\frac{178922400}{9463247}a^{11}+\frac{293167178}{9463247}a^{10}-\frac{130715430}{9463247}a^{9}-\frac{483298085}{9463247}a^{8}-\frac{1798497}{9463247}a^{7}+\frac{367204703}{9463247}a^{6}-\frac{157737075}{9463247}a^{5}-\frac{62289845}{9463247}a^{4}+\frac{121479607}{9463247}a^{3}-\frac{54232040}{9463247}a^{2}-\frac{6939748}{9463247}a+\frac{11883867}{9463247}$, $\frac{21704521}{9463247}a^{15}+\frac{17493406}{9463247}a^{14}-\frac{245897314}{9463247}a^{13}-\frac{282755058}{9463247}a^{12}+\frac{808235869}{9463247}a^{11}+\frac{1320288942}{9463247}a^{10}-\frac{616847019}{9463247}a^{9}-\frac{2226387612}{9463247}a^{8}-\frac{61762133}{9463247}a^{7}+\frac{1646344981}{9463247}a^{6}-\frac{696584014}{9463247}a^{5}-\frac{203208562}{9463247}a^{4}+\frac{683007332}{9463247}a^{3}-\frac{294080097}{9463247}a^{2}-\frac{52850498}{9463247}a+\frac{36720126}{9463247}$, $\frac{5092385}{9463247}a^{15}+\frac{9433346}{9463247}a^{14}-\frac{49778583}{9463247}a^{13}-\frac{122776819}{9463247}a^{12}+\frac{80810335}{9463247}a^{11}+\frac{450518461}{9463247}a^{10}+\frac{291537320}{9463247}a^{9}-\frac{428517082}{9463247}a^{8}-\frac{572427014}{9463247}a^{7}+\frac{24494487}{9463247}a^{6}+\frac{113516749}{9463247}a^{5}-\frac{32826428}{9463247}a^{4}+\frac{54939226}{9463247}a^{3}+\frac{81078385}{9463247}a^{2}-\frac{3329020}{9463247}a-\frac{11700593}{9463247}$, $\frac{11439176}{9463247}a^{15}-\frac{777400}{9463247}a^{14}-\frac{139248888}{9463247}a^{13}-\frac{40222897}{9463247}a^{12}+\frac{572497800}{9463247}a^{11}+\frac{378620476}{9463247}a^{10}-\frac{958596136}{9463247}a^{9}-\frac{1091811819}{9463247}a^{8}+\frac{872674640}{9463247}a^{7}+\frac{1124314589}{9463247}a^{6}-\frac{858819474}{9463247}a^{5}+\frac{110952656}{9463247}a^{4}+\frac{358018135}{9463247}a^{3}-\frac{377529740}{9463247}a^{2}+\frac{40050675}{9463247}a+\frac{16588417}{9463247}$, $\frac{18991077}{9463247}a^{15}+\frac{16200246}{9463247}a^{14}-\frac{214469546}{9463247}a^{13}-\frac{259443016}{9463247}a^{12}+\frac{694526762}{9463247}a^{11}+\frac{1207006304}{9463247}a^{10}-\frac{461574067}{9463247}a^{9}-\frac{2016450663}{9463247}a^{8}-\frac{248640415}{9463247}a^{7}+\frac{1414997388}{9463247}a^{6}-\frac{439304249}{9463247}a^{5}-\frac{139221650}{9463247}a^{4}+\frac{585948237}{9463247}a^{3}-\frac{195171638}{9463247}a^{2}-\frac{77428213}{9463247}a+\frac{12903273}{9463247}$, $\frac{5139798}{9463247}a^{15}+\frac{11210719}{9463247}a^{14}-\frac{50217693}{9463247}a^{13}-\frac{145334724}{9463247}a^{12}+\frac{73308311}{9463247}a^{11}+\frac{545676174}{9463247}a^{10}+\frac{368975249}{9463247}a^{9}-\frac{577093302}{9463247}a^{8}-\frac{796683023}{9463247}a^{7}+\frac{97669152}{9463247}a^{6}+\frac{312732248}{9463247}a^{5}-\frac{84445301}{9463247}a^{4}+\frac{103491386}{9463247}a^{3}+\frac{137093591}{9463247}a^{2}-\frac{53289624}{9463247}a-\frac{23732439}{9463247}$, $\frac{14168626}{9463247}a^{15}+\frac{1651482}{9463247}a^{14}-\frac{175405526}{9463247}a^{13}-\frac{84829214}{9463247}a^{12}+\frac{729545713}{9463247}a^{11}+\frac{645148594}{9463247}a^{10}-\frac{1181367968}{9463247}a^{9}-\frac{1757010862}{9463247}a^{8}+\frac{822417602}{9463247}a^{7}+\frac{1829985074}{9463247}a^{6}-\frac{700741029}{9463247}a^{5}-\frac{153418141}{9463247}a^{4}+\frac{541573624}{9463247}a^{3}-\frac{395271328}{9463247}a^{2}-\frac{110006998}{9463247}a+\frac{40842278}{9463247}$, $\frac{1478752}{9463247}a^{15}+\frac{3327995}{9463247}a^{14}-\frac{14244966}{9463247}a^{13}-\frac{42878288}{9463247}a^{12}+\frac{20309045}{9463247}a^{11}+\frac{156948202}{9463247}a^{10}+\frac{100467049}{9463247}a^{9}-\frac{154477287}{9463247}a^{8}-\frac{195733106}{9463247}a^{7}+\frac{40597106}{9463247}a^{6}+\frac{38792000}{9463247}a^{5}-\frac{80533821}{9463247}a^{4}+\frac{61078472}{9463247}a^{3}+\frac{21248787}{9463247}a^{2}-\frac{1899933}{9463247}a+\frac{4256880}{9463247}$, $\frac{6917373}{9463247}a^{15}+\frac{1045008}{9463247}a^{14}-\frac{83119878}{9463247}a^{13}-\frac{39553202}{9463247}a^{12}+\frac{327955910}{9463247}a^{11}+\frac{266038966}{9463247}a^{10}-\frac{503059018}{9463247}a^{9}-\frac{645662173}{9463247}a^{8}+\frac{445442038}{9463247}a^{7}+\frac{629241628}{9463247}a^{6}-\frac{520838518}{9463247}a^{5}+\frac{47159174}{9463247}a^{4}+\frac{255933627}{9463247}a^{3}-\frac{252107565}{9463247}a^{2}+\frac{24010910}{9463247}a+\frac{28711853}{9463247}$, $\frac{55903296}{9463247}a^{15}+\frac{31114977}{9463247}a^{14}-\frac{652921440}{9463247}a^{13}-\frac{585793773}{9463247}a^{12}+\frac{2350976227}{9463247}a^{11}+\frac{3081414820}{9463247}a^{10}-\frac{2631951594}{9463247}a^{9}-\frac{6097216491}{9463247}a^{8}+\frac{994270801}{9463247}a^{7}+\frac{5166046833}{9463247}a^{6}-\frac{2122028582}{9463247}a^{5}-\frac{471240688}{9463247}a^{4}+\frac{1837663410}{9463247}a^{3}-\frac{1010769502}{9463247}a^{2}-\frac{214900385}{9463247}a+\frac{123667936}{9463247}$, $\frac{34084633}{9463247}a^{15}+\frac{22382059}{9463247}a^{14}-\frac{392211544}{9463247}a^{13}-\frac{392429771}{9463247}a^{12}+\frac{1353524017}{9463247}a^{11}+\frac{1956184115}{9463247}a^{10}-\frac{1286860713}{9463247}a^{9}-\frac{3595272510}{9463247}a^{8}+\frac{210237574}{9463247}a^{7}+\frac{2790973889}{9463247}a^{6}-\frac{1129990346}{9463247}a^{5}-\frac{162162488}{9463247}a^{4}+\frac{1049728014}{9463247}a^{3}-\frac{532570062}{9463247}a^{2}-\frac{99924468}{9463247}a+\frac{51495800}{9463247}$, $\frac{12359979}{9463247}a^{15}+\frac{19820646}{9463247}a^{14}-\frac{128283663}{9463247}a^{13}-\frac{271198824}{9463247}a^{12}+\frac{287643152}{9463247}a^{11}+\frac{1083389326}{9463247}a^{10}+\frac{409392102}{9463247}a^{9}-\frac{1340746244}{9463247}a^{8}-\frac{1227032810}{9463247}a^{7}+\frac{473684905}{9463247}a^{6}+\frac{376797993}{9463247}a^{5}-\frac{93009406}{9463247}a^{4}+\frac{220013424}{9463247}a^{3}+\frac{149173834}{9463247}a^{2}-\frac{56517282}{9463247}a-\frac{44301174}{9463247}$ 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:  \( 52554.9523589 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{2}\cdot 52554.9523589 \cdot 1}{2\cdot\sqrt{673301391207980597248}}\cr\approx \mathstrut & 0.163756512198 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 12*x^14 - 4*x^13 + 48*x^12 + 32*x^11 - 78*x^10 - 84*x^9 + 78*x^8 + 84*x^7 - 88*x^6 + 12*x^5 + 37*x^4 - 36*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 - 12*x^14 - 4*x^13 + 48*x^12 + 32*x^11 - 78*x^10 - 84*x^9 + 78*x^8 + 84*x^7 - 88*x^6 + 12*x^5 + 37*x^4 - 36*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 - 12*x^14 - 4*x^13 + 48*x^12 + 32*x^11 - 78*x^10 - 84*x^9 + 78*x^8 + 84*x^7 - 88*x^6 + 12*x^5 + 37*x^4 - 36*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 - 12*x^14 - 4*x^13 + 48*x^12 + 32*x^11 - 78*x^10 - 84*x^9 + 78*x^8 + 84*x^7 - 88*x^6 + 12*x^5 + 37*x^4 - 36*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

$C_2^4.C_2\wr C_4$ (as 16T1251):

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 1024
The 34 conjugacy class representatives for $C_2^4.C_2\wr C_4$
Character table for $C_2^4.C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.1413480448.1

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

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Arithmetically equvalently siblings: data not computed
Minimal sibling: 16.12.673301391207980597248.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ $16$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $16$ $16$

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$$44$
\(337\) Copy content Toggle raw display $\Q_{337}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{337}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$