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Properties

Label 16.8.396...368.34
Degree $16$
Signature $[8, 4]$
Discriminant $3.966\times 10^{27}$
Root discriminant \(53.08\)
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 - 864*x^10 + 5832*x^8 + 31104*x^6 + 23328*x^4 - 419904*x^2 + 52488)
 
gp: K = bnfinit(y^16 - 144*y^12 - 864*y^10 + 5832*y^8 + 31104*y^6 + 23328*y^4 - 419904*y^2 + 52488, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 144*x^12 - 864*x^10 + 5832*x^8 + 31104*x^6 + 23328*x^4 - 419904*x^2 + 52488);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 144*x^12 - 864*x^10 + 5832*x^8 + 31104*x^6 + 23328*x^4 - 419904*x^2 + 52488)
 

\( x^{16} - 144x^{12} - 864x^{10} + 5832x^{8} + 31104x^{6} + 23328x^{4} - 419904x^{2} + 52488 \) 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:  $[8, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3965881151245791007623610368\) \(\medspace = 2^{79}\cdot 3^{8}\) 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:  \(53.08\)
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:  $2^{2849/512}3^{1/2}\approx 81.96490265902725$
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 \(a\))

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{54}a^{6}$, $\frac{1}{54}a^{7}$, $\frac{1}{162}a^{8}$, $\frac{1}{162}a^{9}$, $\frac{1}{486}a^{10}$, $\frac{1}{972}a^{11}$, $\frac{1}{2916}a^{12}$, $\frac{1}{2916}a^{13}$, $\frac{1}{12573036756}a^{14}-\frac{22621}{4191012252}a^{12}-\frac{33037}{33262002}a^{10}+\frac{104918}{38805669}a^{8}+\frac{71422}{38805669}a^{6}-\frac{8162}{615963}a^{4}-\frac{407756}{4311741}a^{2}-\frac{402818}{1437247}$, $\frac{1}{12573036756}a^{15}-\frac{22621}{4191012252}a^{13}+\frac{7099}{199572012}a^{11}+\frac{104918}{38805669}a^{9}+\frac{71422}{38805669}a^{7}-\frac{8162}{615963}a^{5}-\frac{407756}{4311741}a^{3}-\frac{402818}{1437247}a$ 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:  $11$
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:   $\frac{36691}{6286518378}a^{14}-\frac{46063}{4191012252}a^{12}+\frac{41470}{49893003}a^{10}+\frac{239503}{38805669}a^{8}-\frac{886642}{38805669}a^{6}-\frac{343061}{1847889}a^{4}-\frac{1531748}{4311741}a^{2}+\frac{2605921}{1437247}$, $\frac{88}{19523349}a^{14}+\frac{73}{6507783}a^{12}-\frac{1246}{2169261}a^{10}-\frac{8105}{1446174}a^{8}+\frac{704}{80343}a^{6}+\frac{13016}{80343}a^{4}+\frac{16048}{26781}a^{2}-\frac{4395}{8927}$, $\frac{37963}{6286518378}a^{14}-\frac{11881}{4191012252}a^{12}-\frac{43568}{49893003}a^{10}-\frac{530912}{116417007}a^{8}+\frac{1491088}{38805669}a^{6}+\frac{262940}{1847889}a^{4}-\frac{981676}{4311741}a^{2}-\frac{4055049}{1437247}$, $\frac{39581}{12573036756}a^{14}+\frac{3590}{349251021}a^{12}-\frac{49465}{99786006}a^{10}-\frac{331697}{77611338}a^{8}+\frac{1215913}{77611338}a^{6}+\frac{115132}{615963}a^{4}+\frac{297858}{1437247}a^{2}-\frac{3432781}{1437247}$, $\frac{61501}{12573036756}a^{14}+\frac{40975}{4191012252}a^{12}-\frac{30542}{49893003}a^{10}-\frac{594089}{116417007}a^{8}+\frac{297590}{38805669}a^{6}+\frac{38683}{615963}a^{4}+\frac{1121147}{4311741}a^{2}+\frac{116721}{1437247}$, $\frac{243395}{6286518378}a^{15}+\frac{421417}{6286518378}a^{14}+\frac{127481}{1047753063}a^{13}+\frac{192697}{1047753063}a^{12}-\frac{531073}{99786006}a^{11}-\frac{50794}{5543667}a^{10}-\frac{5803010}{116417007}a^{9}-\frac{6388855}{77611338}a^{8}+\frac{3384944}{38805669}a^{7}+\frac{2140945}{12935223}a^{6}+\frac{2834567}{1847889}a^{5}+\frac{1506244}{615963}a^{4}+\frac{21012400}{4311741}a^{3}+\frac{34064323}{4311741}a^{2}-\frac{2728763}{1437247}a-\frac{782625}{1437247}$, $\frac{67859}{6286518378}a^{15}-\frac{37463}{2095506126}a^{14}-\frac{58643}{2095506126}a^{13}-\frac{69187}{1047753063}a^{12}-\frac{78673}{49893003}a^{11}+\frac{116608}{49893003}a^{10}-\frac{527057}{77611338}a^{9}+\frac{4604161}{232834014}a^{8}+\frac{1110574}{12935223}a^{7}-\frac{45326}{38805669}a^{6}+\frac{750721}{1847889}a^{5}-\frac{536581}{1847889}a^{4}+\frac{1366927}{4311741}a^{3}-\frac{1470670}{1437247}a^{2}-\frac{6838173}{1437247}a+\frac{2375145}{1437247}$, $\frac{278291}{12573036756}a^{15}+\frac{1057}{199572012}a^{14}-\frac{78851}{4191012252}a^{13}-\frac{17165}{598716036}a^{12}-\frac{156704}{49893003}a^{11}-\frac{62377}{99786006}a^{10}-\frac{437527}{25870446}a^{9}-\frac{88003}{33262002}a^{8}+\frac{10882807}{77611338}a^{7}+\frac{75173}{1231926}a^{6}+\frac{1191848}{1847889}a^{5}+\frac{112156}{1847889}a^{4}+\frac{137395}{4311741}a^{3}-\frac{87939}{205321}a^{2}-\frac{14042249}{1437247}a-\frac{923095}{205321}$, $\frac{134819}{12573036756}a^{15}-\frac{14365}{546653772}a^{14}+\frac{97535}{4191012252}a^{13}+\frac{7865}{182217924}a^{12}-\frac{165871}{99786006}a^{11}+\frac{15847}{4338522}a^{10}-\frac{445132}{38805669}a^{9}+\frac{25121}{1687203}a^{8}+\frac{262468}{4311741}a^{7}-\frac{282584}{1687203}a^{6}+\frac{794288}{1847889}a^{5}-\frac{18118}{80343}a^{4}-\frac{2870155}{4311741}a^{3}+\frac{195992}{187467}a^{2}-\frac{4016541}{1437247}a+\frac{61659}{62489}$, $\frac{1424123}{6286518378}a^{15}-\frac{1150495}{12573036756}a^{14}-\frac{172997}{4191012252}a^{13}-\frac{321281}{4191012252}a^{12}+\frac{1618339}{49893003}a^{11}+\frac{1281953}{99786006}a^{10}+\frac{23363188}{116417007}a^{9}+\frac{10162631}{116417007}a^{8}-\frac{98140907}{77611338}a^{7}-\frac{33793493}{77611338}a^{6}-\frac{12989021}{1847889}a^{5}-\frac{552771}{205321}a^{4}-\frac{26908772}{4311741}a^{3}-\frac{10554050}{4311741}a^{2}+\frac{132886039}{1437247}a+\frac{47228911}{1437247}$, $\frac{79441}{598716036}a^{15}+\frac{534337}{12573036756}a^{14}+\frac{4457}{299358018}a^{13}+\frac{3331}{1397004084}a^{12}-\frac{319295}{16631001}a^{11}-\frac{213173}{33262002}a^{10}-\frac{71719}{615963}a^{9}-\frac{8183425}{232834014}a^{8}+\frac{1431835}{1847889}a^{7}+\frac{10258352}{38805669}a^{6}+\frac{865216}{205321}a^{5}+\frac{2563474}{1847889}a^{4}+\frac{2028038}{615963}a^{3}+\frac{1778440}{4311741}a^{2}-\frac{11584146}{205321}a-\frac{28633133}{1437247}$ 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:  \( 149334328.63481638 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 149334328.63481638 \cdot 1}{2\cdot\sqrt{3965881151245791007623610368}}\cr\approx \mathstrut & 0.473063291592085 \end{aligned}\] (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 - 864*x^10 + 5832*x^8 + 31104*x^6 + 23328*x^4 - 419904*x^2 + 52488)
 
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 - 864*x^10 + 5832*x^8 + 31104*x^6 + 23328*x^4 - 419904*x^2 + 52488, 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 - 864*x^10 + 5832*x^8 + 31104*x^6 + 23328*x^4 - 419904*x^2 + 52488);
 
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 - 864*x^10 + 5832*x^8 + 31104*x^6 + 23328*x^4 - 419904*x^2 + 52488);
 
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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.4.2147483648.2

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: 16.8.3965881151245791007623610368.15, 16.4.3965881151245791007623610368.76, 16.8.3965881151245791007623610368.16, 16.8.3965881151245791007623610368.26, 16.8.3965881151245791007623610368.27, 16.4.3965881151245791007623610368.86, 16.4.3965881151245791007623610368.87, 16.4.3965881151245791007623610368.85, 16.12.3965881151245791007623610368.3, 16.8.3965881151245791007623610368.30, 16.4.3965881151245791007623610368.115, 16.0.3965881151245791007623610368.640, some data not computed
Degree 32 siblings: data not computed
Minimal sibling: 16.4.3965881151245791007623610368.76

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 $16$ ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}$ $16$ $16$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ $16$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ $16$ $16$

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 2.16.79.2537$x^{16} + 32 x^{15} + 16 x^{14} + 56 x^{12} + 32 x^{9} + 20 x^{8} + 48 x^{6} + 32 x^{5} + 34$$16$$1$$79$16T1155$[2, 3, 7/2, 4, 17/4, 19/4, 5, 41/8, 43/8, 6]$
\(3\) Copy content Toggle raw display 3.16.8.2$x^{16} - 54 x^{10} + 81 x^{8} + 1458 x^{4} - 4374 x^{2} + 13122$$2$$8$$8$$C_{16}$$[\ ]_{2}^{8}$