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Properties

Label 16.4.321...808.66
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 - 144*x^12 + 6408*x^8 + 10368*x^6 - 95904*x^4 - 378432*x^2 + 187272)
 
gp: K = bnfinit(y^16 - 144*y^12 + 6408*y^8 + 10368*y^6 - 95904*y^4 - 378432*y^2 + 187272, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 144*x^12 + 6408*x^8 + 10368*x^6 - 95904*x^4 - 378432*x^2 + 187272);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 144*x^12 + 6408*x^8 + 10368*x^6 - 95904*x^4 - 378432*x^2 + 187272)
 

\( x^{16} - 144x^{12} + 6408x^{8} + 10368x^{6} - 95904x^{4} - 378432x^{2} + 187272 \) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{6}a^{6}$, $\frac{1}{6}a^{7}$, $\frac{1}{18}a^{8}$, $\frac{1}{18}a^{9}$, $\frac{1}{18}a^{10}$, $\frac{1}{36}a^{11}$, $\frac{1}{756}a^{12}+\frac{1}{126}a^{10}+\frac{1}{126}a^{8}-\frac{1}{21}a^{6}+\frac{2}{21}a^{4}-\frac{1}{7}a^{2}+\frac{1}{7}$, $\frac{1}{756}a^{13}+\frac{1}{126}a^{11}+\frac{1}{126}a^{9}-\frac{1}{21}a^{7}+\frac{2}{21}a^{5}-\frac{1}{7}a^{3}+\frac{1}{7}a$, $\frac{1}{1709605010484}a^{14}-\frac{79858843}{244229287212}a^{12}-\frac{2306461387}{142467084207}a^{10}-\frac{1686632519}{284934168414}a^{8}-\frac{614295035}{31659352046}a^{6}+\frac{6086962328}{47489028069}a^{4}-\frac{447831428}{2261382289}a^{2}-\frac{7660378998}{15829676023}$, $\frac{1}{29063285178228}a^{15}-\frac{4697579}{244229287212}a^{13}+\frac{2201276825}{3229253908692}a^{11}+\frac{253142966}{15829676023}a^{9}+\frac{6993395459}{807313477173}a^{7}-\frac{24353805929}{269104492391}a^{5}-\frac{447831428}{38443498913}a^{3}-\frac{118468111159}{269104492391}a$ 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:   $\frac{3030655}{854802505242}a^{14}+\frac{11544871}{1709605010484}a^{12}+\frac{9809232}{15829676023}a^{10}-\frac{37532729}{47489028069}a^{8}-\frac{570855660}{15829676023}a^{6}-\frac{35645011}{6784146867}a^{4}+\frac{13926157068}{15829676023}a^{2}+\frac{7258801031}{15829676023}$, $\frac{40}{84449961}a^{14}-\frac{337}{9383329}a^{12}-\frac{10366}{84449961}a^{10}+\frac{633671}{168899922}a^{8}+\frac{56848}{9383329}a^{6}-\frac{511088}{9383329}a^{4}-\frac{2959640}{9383329}a^{2}-\frac{21233771}{9383329}$, $\frac{2286761}{854802505242}a^{14}+\frac{8371697}{1709605010484}a^{12}-\frac{7718342}{47489028069}a^{10}-\frac{9190610}{47489028069}a^{8}-\frac{125810014}{15829676023}a^{6}+\frac{905194844}{47489028069}a^{4}+\frac{4540043244}{15829676023}a^{2}+\frac{3695077135}{2261382289}$, $\frac{1977793}{81409762404}a^{14}+\frac{13448389}{189956112276}a^{12}-\frac{390206689}{94978056138}a^{10}-\frac{246944249}{47489028069}a^{8}+\frac{8975837914}{47489028069}a^{6}+\frac{22192452371}{47489028069}a^{4}-\frac{40846513497}{15829676023}a^{2}-\frac{212163717697}{15829676023}$, $\frac{157303535}{29063285178228}a^{15}+\frac{42511073}{854802505242}a^{14}-\frac{52680034}{427401252621}a^{13}+\frac{162648359}{569868336828}a^{12}-\frac{1657293721}{1614626954346}a^{11}-\frac{612550327}{94978056138}a^{10}+\frac{1673435488}{142467084207}a^{9}-\frac{2186931079}{94978056138}a^{8}+\frac{13598953763}{269104492391}a^{7}+\frac{10811150765}{47489028069}a^{6}-\frac{347283438}{38443498913}a^{5}+\frac{11595121589}{47489028069}a^{4}-\frac{238705027255}{269104492391}a^{3}-\frac{3951076777}{15829676023}a^{2}-\frac{2286017387242}{269104492391}a-\frac{56280754121}{15829676023}$, $\frac{364481659}{29063285178228}a^{15}-\frac{5165245}{284934168414}a^{14}+\frac{28320538}{427401252621}a^{13}-\frac{74350877}{189956112276}a^{12}+\frac{10267945211}{4843880863038}a^{11}+\frac{468269477}{284934168414}a^{10}-\frac{1485256807}{142467084207}a^{9}+\frac{4665776765}{94978056138}a^{8}-\frac{91617019271}{807313477173}a^{7}-\frac{698990720}{47489028069}a^{6}+\frac{70558257198}{269104492391}a^{5}-\frac{66938737415}{47489028069}a^{4}+\frac{433064814065}{269104492391}a^{3}-\frac{23247871285}{15829676023}a^{2}-\frac{226239399062}{269104492391}a+\frac{14345669075}{15829676023}$, $\frac{38708758}{7265821294557}a^{15}+\frac{65631472}{142467084207}a^{14}+\frac{153251033}{427401252621}a^{13}-\frac{56834121}{31659352046}a^{12}-\frac{273331557}{76886997826}a^{11}-\frac{1339696118}{20352440601}a^{10}-\frac{221320999}{6784146867}a^{9}+\frac{1891348313}{6784146867}a^{8}+\frac{431234975242}{807313477173}a^{7}+\frac{42468474803}{15829676023}a^{6}-\frac{150505502342}{269104492391}a^{5}-\frac{143154151864}{15829676023}a^{4}-\frac{3972304742480}{269104492391}a^{3}-\frac{525994644464}{15829676023}a^{2}+\frac{12128670497820}{269104492391}a+\frac{666821741461}{15829676023}$, $\frac{4007549031209}{29063285178228}a^{15}+\frac{13142534023}{142467084207}a^{14}-\frac{52699348739}{854802505242}a^{13}+\frac{70306985809}{1709605010484}a^{12}+\frac{32015158972315}{1614626954346}a^{11}-\frac{1889954122970}{142467084207}a^{10}+\frac{842038898195}{94978056138}a^{9}-\frac{280863972955}{47489028069}a^{8}-\frac{710099443506941}{807313477173}a^{7}+\frac{27947659604638}{47489028069}a^{6}-\frac{210250948659490}{115330496739}a^{5}+\frac{2756061486978}{2261382289}a^{4}+\frac{33\!\cdots\!86}{269104492391}a^{3}-\frac{131418534284573}{15829676023}a^{2}+\frac{15\!\cdots\!16}{269104492391}a-\frac{611180056567579}{15829676023}$, $\frac{1237870556323}{14531642589114}a^{15}+\frac{357817845865}{1709605010484}a^{14}+\frac{219763679908}{427401252621}a^{13}+\frac{239876193541}{189956112276}a^{12}-\frac{44399672956315}{4843880863038}a^{11}-\frac{3209640426767}{142467084207}a^{10}-\frac{5258946327095}{94978056138}a^{9}-\frac{6460587498929}{47489028069}a^{8}+\frac{56965174799625}{269104492391}a^{7}+\frac{8239472007877}{15829676023}a^{6}+\frac{17\!\cdots\!47}{807313477173}a^{5}+\frac{252512931357605}{47489028069}a^{4}+\frac{13\!\cdots\!30}{269104492391}a^{3}+\frac{190631731064807}{15829676023}a^{2}-\frac{711262547944454}{269104492391}a-\frac{14675151903575}{2261382289}$ 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:  \( 543718658.5656872 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 543718658.5656872 \cdot 1}{2\cdot\sqrt{321236373250909071617512439808}}\cr\approx \mathstrut & 0.472205538762194 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 144*x^12 + 6408*x^8 + 10368*x^6 - 95904*x^4 - 378432*x^2 + 187272)
 
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 - 144*x^12 + 6408*x^8 + 10368*x^6 - 95904*x^4 - 378432*x^2 + 187272, 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 - 144*x^12 + 6408*x^8 + 10368*x^6 - 95904*x^4 - 378432*x^2 + 187272);
 
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 - 144*x^12 + 6408*x^8 + 10368*x^6 - 95904*x^4 - 378432*x^2 + 187272);
 
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 $16$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ $16$ $16$

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 2.16.79.2543$x^{16} + 16 x^{14} + 24 x^{12} + 32 x^{11} + 32 x^{9} + 52 x^{8} + 32 x^{7} + 48 x^{6} + 32 x^{5} + 32 x^{2} + 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}$