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Properties

Label 16.0.269...625.4
Degree $16$
Signature $[0, 8]$
Discriminant $2.697\times 10^{29}$
Root discriminant \(69.09\)
Ramified primes $3,5,17$
Class number $35040$ (GRH)
Class group [2, 17520] (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536)
 
gp: K = bnfinit(y^16 - y^15 - 15*y^14 + 47*y^13 + 255*y^12 - 1999*y^11 + 4971*y^10 - 6331*y^9 - 3164*y^8 + 22124*y^7 - 12352*y^6 - 29264*y^5 + 196928*y^4 - 293824*y^3 + 335616*y^2 - 212992*y + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536)
 

\( x^{16} - x^{15} - 15 x^{14} + 47 x^{13} + 255 x^{12} - 1999 x^{11} + 4971 x^{10} - 6331 x^{9} + \cdots + 65536 \) 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:   \(269708720716852904093994140625\) \(\medspace = 3^{8}\cdot 5^{12}\cdot 17^{14}\) 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:  \(69.09\)
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:  $3^{1/2}5^{3/4}17^{7/8}\approx 69.09250471045908$
Ramified primes:   \(3\), \(5\), \(17\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(64,·)$, $\chi_{255}(1,·)$, $\chi_{255}(2,·)$, $\chi_{255}(4,·)$, $\chi_{255}(8,·)$, $\chi_{255}(128,·)$, $\chi_{255}(16,·)$, $\chi_{255}(223,·)$, $\chi_{255}(32,·)$, $\chi_{255}(127,·)$, $\chi_{255}(239,·)$, $\chi_{255}(247,·)$, $\chi_{255}(251,·)$, $\chi_{255}(253,·)$, $\chi_{255}(254,·)$, $\chi_{255}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{16}a^{7}+\frac{3}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{128}a^{8}-\frac{1}{32}a^{7}+\frac{1}{64}a^{6}-\frac{1}{16}a^{5}+\frac{1}{128}a^{4}-\frac{1}{32}a^{3}-\frac{1}{32}a^{2}+\frac{1}{8}a$, $\frac{1}{128}a^{9}+\frac{1}{64}a^{7}+\frac{1}{128}a^{5}-\frac{1}{32}a^{3}$, $\frac{1}{1024}a^{10}-\frac{3}{1024}a^{9}-\frac{1}{256}a^{8}-\frac{15}{512}a^{7}+\frac{21}{1024}a^{6}-\frac{19}{1024}a^{5}+\frac{11}{512}a^{4}-\frac{43}{256}a^{3}-\frac{21}{128}a^{2}-\frac{5}{32}a-\frac{1}{2}$, $\frac{1}{2048}a^{11}-\frac{1}{2048}a^{10}+\frac{3}{1024}a^{9}-\frac{3}{1024}a^{8}-\frac{7}{2048}a^{7}-\frac{41}{2048}a^{6}-\frac{1}{16}a^{5}+\frac{1}{64}a^{4}+\frac{1}{32}a^{3}+\frac{25}{128}a^{2}+\frac{11}{32}a-\frac{1}{2}$, $\frac{1}{2048}a^{12}-\frac{1}{2048}a^{10}+\frac{1}{1024}a^{9}-\frac{5}{2048}a^{8}+\frac{9}{512}a^{7}+\frac{57}{2048}a^{6}+\frac{1}{1024}a^{5}+\frac{19}{512}a^{4}-\frac{21}{256}a^{3}-\frac{3}{16}a^{2}+\frac{3}{16}a$, $\frac{1}{4096}a^{13}-\frac{1}{4096}a^{11}-\frac{1}{2048}a^{10}+\frac{7}{4096}a^{9}-\frac{3}{1024}a^{8}+\frac{49}{4096}a^{7}-\frac{41}{2048}a^{6}-\frac{29}{512}a^{5}+\frac{1}{64}a^{4}+\frac{51}{256}a^{3}+\frac{25}{128}a^{2}+\frac{5}{32}a-\frac{1}{2}$, $\frac{1}{1612693504}a^{14}+\frac{82403}{1612693504}a^{13}+\frac{160433}{1612693504}a^{12}-\frac{108489}{1612693504}a^{11}+\frac{106367}{1612693504}a^{10}+\frac{3205841}{1612693504}a^{9}-\frac{6060869}{1612693504}a^{8}-\frac{23516035}{1612693504}a^{7}-\frac{11028959}{403173376}a^{6}-\frac{2963847}{201586688}a^{5}-\frac{1159779}{100793344}a^{4}+\frac{16689133}{100793344}a^{3}-\frac{4696869}{25198336}a^{2}+\frac{625623}{1574896}a+\frac{15704}{98431}$, $\frac{1}{103212384256}a^{15}+\frac{21}{103212384256}a^{14}-\frac{2924097}{103212384256}a^{13}-\frac{8247143}{103212384256}a^{12}+\frac{19798565}{103212384256}a^{11}-\frac{34596513}{103212384256}a^{10}+\frac{119309717}{103212384256}a^{9}-\frac{174798829}{103212384256}a^{8}-\frac{1084955485}{51606192128}a^{7}-\frac{47282041}{6450774016}a^{6}+\frac{163332395}{3225387008}a^{5}-\frac{208610497}{6450774016}a^{4}+\frac{239072703}{3225387008}a^{3}+\frac{6424291}{806346752}a^{2}+\frac{5810793}{12599168}a-\frac{73363}{3149792}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{2}\times C_{17520}$, which has order $35040$ (assuming GRH)

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:   \( -1 \)  (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:   $\frac{297593}{51606192128}a^{15}-\frac{2313235}{51606192128}a^{14}-\frac{4036025}{51606192128}a^{13}+\frac{51057745}{51606192128}a^{12}+\frac{9835645}{51606192128}a^{11}-\frac{1271475225}{51606192128}a^{10}+\frac{5184337005}{51606192128}a^{9}-\frac{6925233925}{51606192128}a^{8}-\frac{1854061685}{25803096064}a^{7}+\frac{1796565135}{3225387008}a^{6}-\frac{1001860013}{1612693504}a^{5}-\frac{4534604345}{3225387008}a^{4}+\frac{5403534535}{1612693504}a^{3}-\frac{1954563765}{403173376}a^{2}+\frac{21086505}{6299584}a-\frac{1447947}{1574896}$, $\frac{3006735}{51606192128}a^{15}-\frac{12591941}{51606192128}a^{14}-\frac{45669967}{51606192128}a^{13}+\frac{306716791}{51606192128}a^{12}+\frac{482566571}{51606192128}a^{11}-\frac{9134316783}{51606192128}a^{10}+\frac{31622866491}{51606192128}a^{9}-\frac{42195869795}{51606192128}a^{8}-\frac{7506139315}{25803096064}a^{7}+\frac{10285373545}{3225387008}a^{6}-\frac{5322203515}{1612693504}a^{5}-\frac{20316112975}{3225387008}a^{4}+\frac{32756687217}{1612693504}a^{3}-\frac{15049600371}{403173376}a^{2}+\frac{177102495}{6299584}a-\frac{13977613}{1574896}$, $\frac{1232941}{25803096064}a^{15}-\frac{1873039}{25803096064}a^{14}-\frac{19652525}{25803096064}a^{13}+\frac{64278181}{25803096064}a^{12}+\frac{309380673}{25803096064}a^{11}-\frac{2627464973}{25803096064}a^{10}+\frac{6829529713}{25803096064}a^{9}-\frac{9039820137}{25803096064}a^{8}-\frac{290528441}{12901548032}a^{7}+\frac{1892951515}{1612693504}a^{6}-\frac{803865073}{806346752}a^{5}-\frac{1394159085}{1612693504}a^{4}+\frac{7029066611}{806346752}a^{3}-\frac{4588458617}{201586688}a^{2}+\frac{58967933}{3149792}a-\frac{15716239}{787448}$, $\frac{31045963}{25803096064}a^{15}-\frac{11832857}{25803096064}a^{14}-\frac{463782603}{25803096064}a^{13}+\frac{1161680531}{25803096064}a^{12}+\frac{8465079063}{25803096064}a^{11}-\frac{56354737643}{25803096064}a^{10}+\frac{122332525927}{25803096064}a^{9}-\frac{140913934639}{25803096064}a^{8}-\frac{73813494623}{12901548032}a^{7}+\frac{37196941613}{1612693504}a^{6}-\frac{4500138135}{806346752}a^{5}-\frac{49317649995}{1612693504}a^{4}+\frac{183289720021}{806346752}a^{3}-\frac{49228634367}{201586688}a^{2}+\frac{921874029}{3149792}a-\frac{88146993}{787448}$, $\frac{1149667}{6450774016}a^{15}-\frac{701897}{6450774016}a^{14}-\frac{20640091}{6450774016}a^{13}+\frac{47926659}{6450774016}a^{12}+\frac{366585143}{6450774016}a^{11}-\frac{2287527131}{6450774016}a^{10}+\frac{3841716791}{6450774016}a^{9}+\frac{403379361}{6450774016}a^{8}-\frac{7170327803}{3225387008}a^{7}+\frac{1279868395}{403173376}a^{6}+\frac{924221263}{201586688}a^{5}-\frac{4497998795}{403173376}a^{4}+\frac{3485880889}{201586688}a^{3}-\frac{544811715}{50396672}a^{2}+\frac{1871273}{787448}a+\frac{125885}{196862}$, $\frac{75625267}{51606192128}a^{15}-\frac{245836881}{51606192128}a^{14}-\frac{1166199155}{51606192128}a^{13}+\frac{6407863803}{51606192128}a^{12}+\frac{14527479519}{51606192128}a^{11}-\frac{205807124883}{51606192128}a^{10}+\frac{664186748207}{51606192128}a^{9}-\frac{882086009271}{51606192128}a^{8}-\frac{137117939911}{25803096064}a^{7}+\frac{209873996229}{3225387008}a^{6}-\frac{105040703791}{1612693504}a^{5}-\frac{368705459315}{3225387008}a^{4}+\frac{689223100749}{1612693504}a^{3}-\frac{343935856583}{403173376}a^{2}+\frac{4147623719}{6299584}a-\frac{363819945}{1574896}$, $\frac{31060379}{25803096064}a^{15}-\frac{11226345}{25803096064}a^{14}-\frac{475841083}{25803096064}a^{13}+\frac{4508675}{100401152}a^{12}+\frac{8714692615}{25803096064}a^{11}-\frac{56617111259}{25803096064}a^{10}+\frac{117442374551}{25803096064}a^{9}-\frac{116038824031}{25803096064}a^{8}-\frac{90986276575}{12901548032}a^{7}+\frac{36151423173}{1612693504}a^{6}+\frac{69094657}{806346752}a^{5}-\frac{60831497595}{1612693504}a^{4}+\frac{170170891029}{806346752}a^{3}-\frac{43835444383}{201586688}a^{2}+\frac{763695921}{3149792}a-\frac{70812321}{787448}$ Copy content Toggle raw display (assuming GRH)
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:  \( 101041909.11303878 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 101041909.11303878 \cdot 35040}{2\cdot\sqrt{269708720716852904093994140625}}\cr\approx \mathstrut & 8279.93824111437 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536)
 
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 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536, 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 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536);
 
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 - x^15 - 15*x^14 + 47*x^13 + 255*x^12 - 1999*x^11 + 4971*x^10 - 6331*x^9 - 3164*x^8 + 22124*x^7 - 12352*x^6 - 29264*x^5 + 196928*x^4 - 293824*x^3 + 335616*x^2 - 212992*x + 65536);
 
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

$C_2\times C_8$ (as 16T5):

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)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{17})\), 4.4.122825.1, 4.0.44217.1, 8.0.1221964430625.1, 8.8.519334883015625.2, 8.0.6411541765625.2

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{16}$ R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(3\) Copy content Toggle raw display 3.16.8.1$x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(5\) Copy content Toggle raw display 5.16.12.2$x^{16} - 40 x^{12} + 500 x^{8} + 2500$$4$$4$$12$$C_8\times C_2$$[\ ]_{4}^{4}$
\(17\) Copy content Toggle raw display 17.8.7.2$x^{8} + 136$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.2$x^{8} + 136$$8$$1$$7$$C_8$$[\ ]_{8}$