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Properties

Label 16.4.321...808.61
Degree $16$
Signature $[4, 6]$
Discriminant $3.212\times 10^{29}$
Root discriminant \(69.85\)
Ramified primes $2,3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^7.C_8$ (as 16T1155)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 264*x^12 - 432*x^10 + 11268*x^8 + 60912*x^6 + 122256*x^4 + 119232*x^2 + 46818)
 
gp: K = bnfinit(y^16 - 264*y^12 - 432*y^10 + 11268*y^8 + 60912*y^6 + 122256*y^4 + 119232*y^2 + 46818, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 264*x^12 - 432*x^10 + 11268*x^8 + 60912*x^6 + 122256*x^4 + 119232*x^2 + 46818);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 264*x^12 - 432*x^10 + 11268*x^8 + 60912*x^6 + 122256*x^4 + 119232*x^2 + 46818)
 

\( x^{16} - 264x^{12} - 432x^{10} + 11268x^{8} + 60912x^{6} + 122256x^{4} + 119232x^{2} + 46818 \) 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:  $[4, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(321236373250909071617512439808\) \(\medspace = 2^{79}\cdot 3^{12}\) 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:  \(69.85\)
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:  $2^{2849/512}3^{3/4}\approx 107.8718783637264$
Ramified primes:   \(2\), \(3\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{9}a^{8}$, $\frac{1}{9}a^{9}$, $\frac{1}{9}a^{10}$, $\frac{1}{9}a^{11}$, $\frac{1}{27}a^{12}$, $\frac{1}{459}a^{13}+\frac{4}{153}a^{11}-\frac{2}{51}a^{9}-\frac{4}{51}a^{7}-\frac{1}{17}a^{5}+\frac{6}{17}a$, $\frac{1}{26553355647147}a^{14}-\frac{447245433917}{26553355647147}a^{12}-\frac{95685526241}{2950372849683}a^{10}+\frac{22318412882}{983457616561}a^{8}+\frac{37910431289}{983457616561}a^{6}+\frac{7402776417}{57850448033}a^{4}+\frac{19327096081}{140493945223}a^{2}-\frac{15580865207}{57850448033}$, $\frac{1}{26553355647147}a^{15}+\frac{15558150347}{26553355647147}a^{13}-\frac{402757474789}{8851118549049}a^{11}+\frac{124805686679}{2950372849683}a^{9}+\frac{229432189933}{2950372849683}a^{7}-\frac{27411538964}{2950372849683}a^{5}+\frac{19327096081}{140493945223}a^{3}-\frac{438426052618}{983457616561}a$ 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 $1$ (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:  $9$
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:   $\frac{4077839846}{26553355647147}a^{14}+\frac{5198516693}{26553355647147}a^{12}+\frac{118793685386}{2950372849683}a^{10}+\frac{133667739866}{8851118549049}a^{8}-\frac{5136378776164}{2950372849683}a^{6}-\frac{1241713079189}{173551344099}a^{4}-\frac{1442059862232}{140493945223}a^{2}-\frac{267366723413}{57850448033}$, $\frac{484804}{9701627931}a^{14}+\frac{1416088}{9701627931}a^{12}+\frac{41418376}{3233875977}a^{10}-\frac{52666361}{3233875977}a^{8}-\frac{565718536}{1077958659}a^{6}-\frac{1526461204}{1077958659}a^{4}-\frac{589197432}{359319553}a^{2}-\frac{1128915099}{359319553}$, $\frac{7403343998}{26553355647147}a^{14}-\frac{5299232878}{26553355647147}a^{12}-\frac{651584477738}{8851118549049}a^{10}-\frac{596682814988}{8851118549049}a^{8}+\frac{9529463109224}{2950372849683}a^{6}+\frac{2542618499977}{173551344099}a^{4}+\frac{3054993313380}{140493945223}a^{2}+\frac{649398462925}{57850448033}$, $\frac{84089086}{8851118549049}a^{14}+\frac{302968324}{8851118549049}a^{12}+\frac{19856342972}{8851118549049}a^{10}-\frac{61656679268}{8851118549049}a^{8}-\frac{186926466289}{2950372849683}a^{6}+\frac{14460102869}{57850448033}a^{4}+\frac{161224754392}{140493945223}a^{2}+\frac{69811032225}{57850448033}$, $\frac{14304933374}{26553355647147}a^{15}+\frac{34962609628}{26553355647147}a^{14}+\frac{78948732}{57850448033}a^{13}-\frac{27277384084}{8851118549049}a^{12}+\frac{1242458688148}{8851118549049}a^{11}-\frac{2985771819796}{8851118549049}a^{10}-\frac{1144367600120}{8851118549049}a^{9}+\frac{608437696766}{2950372849683}a^{8}-\frac{6044912649984}{983457616561}a^{7}+\frac{40073481647588}{2950372849683}a^{6}-\frac{48254508899332}{2950372849683}a^{5}+\frac{2934751975458}{57850448033}a^{4}-\frac{1239542813716}{140493945223}a^{3}+\frac{10231024472506}{140493945223}a^{2}+\frac{871848902558}{983457616561}a+\frac{2042858393195}{57850448033}$, $\frac{1782510949}{26553355647147}a^{14}-\frac{2759228636}{26553355647147}a^{12}+\frac{163280552144}{8851118549049}a^{10}+\frac{148041968189}{2950372849683}a^{8}-\frac{2422527021947}{2950372849683}a^{6}-\frac{288893735559}{57850448033}a^{4}-\frac{1369412726992}{140493945223}a^{2}-\frac{328416306089}{57850448033}$, $\frac{19496791067}{26553355647147}a^{15}+\frac{2542565572}{2950372849683}a^{14}+\frac{99724209}{983457616561}a^{13}-\frac{33629966521}{26553355647147}a^{12}+\frac{1714708095818}{8851118549049}a^{11}-\frac{226187338949}{983457616561}a^{10}+\frac{2500554351202}{8851118549049}a^{9}-\frac{53392990337}{983457616561}a^{8}-\frac{8213717673624}{983457616561}a^{7}+\frac{10520507390532}{983457616561}a^{6}-\frac{124043519617720}{2950372849683}a^{5}+\frac{7261326878287}{173551344099}a^{4}-\frac{9830703707746}{140493945223}a^{3}+\frac{6601128709236}{140493945223}a^{2}-\frac{36737924380754}{983457616561}a+\frac{814332648637}{57850448033}$, $\frac{33714606197}{1561962096891}a^{15}-\frac{70920836894}{3793336521021}a^{14}+\frac{369106132235}{26553355647147}a^{13}+\frac{14510666173}{421481835669}a^{12}+\frac{50498446841737}{8851118549049}a^{11}+\frac{688370036671}{140493945223}a^{10}+\frac{49796711309839}{8851118549049}a^{9}-\frac{420206749517}{421481835669}a^{8}-\frac{740678265463760}{2950372849683}a^{7}-\frac{91036577362156}{421481835669}a^{6}-\frac{11\!\cdots\!44}{983457616561}a^{5}-\frac{6109055067604}{8264349719}a^{4}-\frac{14084630299202}{8264349719}a^{3}-\frac{84731303873392}{140493945223}a^{2}-\frac{792602213726634}{983457616561}a-\frac{543666042217}{8264349719}$, $\frac{225031972}{1154493723789}a^{15}-\frac{393100584040}{26553355647147}a^{14}-\frac{8524271453}{384831241263}a^{13}+\frac{301175135816}{8851118549049}a^{12}+\frac{17895952516}{384831241263}a^{11}+\frac{3685492632959}{983457616561}a^{10}+\frac{2099634062543}{384831241263}a^{9}-\frac{16381249763204}{8851118549049}a^{8}-\frac{1705939565308}{128277080421}a^{7}-\frac{139788694117585}{983457616561}a^{6}-\frac{25701163279336}{128277080421}a^{5}-\frac{109607461364627}{173551344099}a^{4}-\frac{1823223790714}{6108432401}a^{3}-\frac{154525927463676}{140493945223}a^{2}-\frac{4427660756628}{42759026807}a-\frac{44254080424385}{57850448033}$ 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:  \( 384167374.447405 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 384167374.447405 \cdot 1}{2\cdot\sqrt{321236373250909071617512439808}}\cr\approx \mathstrut & 0.333639390828223 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 264*x^12 - 432*x^10 + 11268*x^8 + 60912*x^6 + 122256*x^4 + 119232*x^2 + 46818)
 
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 - 264*x^12 - 432*x^10 + 11268*x^8 + 60912*x^6 + 122256*x^4 + 119232*x^2 + 46818, 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 - 264*x^12 - 432*x^10 + 11268*x^8 + 60912*x^6 + 122256*x^4 + 119232*x^2 + 46818);
 
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 - 264*x^12 - 432*x^10 + 11268*x^8 + 60912*x^6 + 122256*x^4 + 119232*x^2 + 46818);
 
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

$C_2^7.C_8$ (as 16T1155):

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 40 conjugacy class representatives for $C_2^7.C_8$
Character table for $C_2^7.C_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.173946175488.2

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

Degree 16 siblings: 16.12.321236373250909071617512439808.3, 16.8.321236373250909071617512439808.4, 16.4.321236373250909071617512439808.20, 16.8.321236373250909071617512439808.9, 16.12.321236373250909071617512439808.6, 16.8.321236373250909071617512439808.15, 16.4.321236373250909071617512439808.42, 16.8.321236373250909071617512439808.18, some data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.8.321236373250909071617512439808.19

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 $16$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ $16$ $16$

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 2.16.79.2534$x^{16} + 40 x^{12} + 32 x^{10} + 36 x^{8} + 32 x^{7} + 16 x^{6} + 32 x^{5} + 2$$16$$1$$79$16T1155$[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$
\(3\) Copy content Toggle raw display 3.16.12.3$x^{16} - 6 x^{12} + 162$$4$$4$$12$$C_{16} : C_2$$[\ ]_{4}^{8}$