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Properties

Label 16.0.104...369.7
Degree $16$
Signature $[0, 8]$
Discriminant $1.048\times 10^{31}$
Root discriminant \(86.85\)
Ramified primes $17,53$
Class number $484$ (GRH)
Class group [11, 44] (GRH)
Galois group $QD_{16}$ (as 16T12)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008)
 
gp: K = bnfinit(y^16 - 8*y^15 + 132*y^14 - 784*y^13 + 6578*y^12 - 29640*y^11 + 161878*y^10 - 555422*y^9 + 2130764*y^8 - 5486248*y^7 + 15012390*y^6 - 27873848*y^5 + 52839433*y^4 - 64659186*y^3 + 76467296*y^2 - 48013336*y + 22193008, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008)
 

\( x^{16} - 8 x^{15} + 132 x^{14} - 784 x^{13} + 6578 x^{12} - 29640 x^{11} + 161878 x^{10} + \cdots + 22193008 \) 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:   \(10483151353726139536553735554369\) \(\medspace = 17^{14}\cdot 53^{8}\) 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:  \(86.85\)
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:  $17^{7/8}53^{1/2}\approx 86.8521834484431$
Ramified primes:   \(17\), \(53\) 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 over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{204}a^{8}-\frac{1}{51}a^{7}-\frac{11}{51}a^{6}+\frac{11}{51}a^{5}-\frac{1}{102}a^{4}-\frac{10}{51}a^{3}-\frac{41}{204}a^{2}+\frac{43}{102}a-\frac{4}{51}$, $\frac{1}{204}a^{9}+\frac{7}{34}a^{7}-\frac{5}{34}a^{6}-\frac{5}{34}a^{5}-\frac{4}{17}a^{4}-\frac{33}{68}a^{3}-\frac{13}{34}a^{2}+\frac{11}{102}a-\frac{16}{51}$, $\frac{1}{204}a^{10}+\frac{3}{17}a^{7}-\frac{3}{34}a^{6}+\frac{7}{34}a^{5}-\frac{5}{68}a^{4}-\frac{5}{34}a^{3}+\frac{5}{102}a^{2}+\frac{49}{102}a+\frac{5}{17}$, $\frac{1}{204}a^{11}+\frac{2}{17}a^{7}-\frac{1}{34}a^{6}+\frac{11}{68}a^{5}+\frac{7}{34}a^{4}+\frac{11}{102}a^{3}-\frac{29}{102}a^{2}-\frac{13}{34}a-\frac{3}{17}$, $\frac{1}{12758160}a^{12}-\frac{1}{2126360}a^{11}+\frac{551}{425272}a^{10}-\frac{1337}{850544}a^{9}-\frac{6877}{3189540}a^{8}-\frac{196679}{797385}a^{7}+\frac{39223}{2551632}a^{6}+\frac{1099403}{6379080}a^{5}-\frac{3139}{2126360}a^{4}+\frac{1061969}{4252720}a^{3}-\frac{219193}{3189540}a^{2}-\frac{1183}{6018}a+\frac{8491}{30090}$, $\frac{1}{12758160}a^{13}+\frac{2749}{2126360}a^{11}+\frac{3317}{2551632}a^{10}-\frac{3793}{2126360}a^{9}+\frac{677}{3189540}a^{8}+\frac{2953771}{12758160}a^{7}+\frac{203047}{1594770}a^{6}-\frac{229859}{6379080}a^{5}-\frac{131941}{750480}a^{4}-\frac{6187}{21624}a^{3}-\frac{79311}{1063180}a^{2}+\frac{1817}{10030}a-\frac{341}{885}$, $\frac{1}{12758160}a^{14}-\frac{3177}{4252720}a^{11}+\frac{2911}{6379080}a^{10}+\frac{7909}{6379080}a^{9}+\frac{3643}{12758160}a^{8}-\frac{82828}{797385}a^{7}+\frac{8672}{797385}a^{6}-\frac{2027921}{12758160}a^{5}+\frac{776153}{6379080}a^{4}-\frac{363157}{1275816}a^{3}+\frac{163133}{531590}a^{2}-\frac{13459}{30090}a+\frac{1591}{5015}$, $\frac{1}{661395772560}a^{15}+\frac{25913}{661395772560}a^{14}-\frac{802}{41337235785}a^{13}-\frac{30}{2755815719}a^{12}+\frac{44192923}{44093051504}a^{11}+\frac{66472007}{330697886280}a^{10}+\frac{168796571}{82674471570}a^{9}+\frac{504559149}{220465257520}a^{8}+\frac{9565133767}{41337235785}a^{7}+\frac{15913674461}{82674471570}a^{6}+\frac{8867416457}{44093051504}a^{5}-\frac{67518294871}{330697886280}a^{4}-\frac{6712283867}{220465257520}a^{3}+\frac{75437128543}{165348943140}a^{2}+\frac{146491089}{519965230}a-\frac{45562199}{91758570}$ 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

$C_{11}\times C_{44}$, which has order $484$ (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:   $\frac{9}{425272}a^{14}-\frac{63}{425272}a^{13}+\frac{2185}{850544}a^{12}-\frac{717}{53159}a^{11}+\frac{24717}{212636}a^{10}-\frac{392183}{850544}a^{9}+\frac{1073439}{425272}a^{8}-\frac{3177729}{425272}a^{7}+\frac{23607771}{850544}a^{6}-\frac{6263199}{106318}a^{5}+\frac{31187955}{212636}a^{4}-\frac{172685021}{850544}a^{3}+\frac{16089096}{53159}a^{2}-\frac{420951}{2006}a+\frac{201735}{2006}$, $\frac{3}{106318}a^{14}-\frac{21}{106318}a^{13}+\frac{10807}{3189540}a^{12}-\frac{4721}{265795}a^{11}+\frac{24127}{159477}a^{10}-\frac{127227}{212636}a^{9}+\frac{2580436}{797385}a^{8}-\frac{16889}{1770}a^{7}+\frac{22356617}{637908}a^{6}-\frac{59062372}{797385}a^{5}+\frac{145863578}{797385}a^{4}-\frac{47361583}{187620}a^{3}+\frac{200239091}{531590}a^{2}-\frac{262167}{1003}a+\frac{2023309}{15045}$, $\frac{41}{3189540}a^{14}-\frac{287}{3189540}a^{13}+\frac{5853}{4252720}a^{12}-\frac{9043}{1275816}a^{11}+\frac{339679}{6379080}a^{10}-\frac{2595209}{12758160}a^{9}+\frac{3089159}{3189540}a^{8}-\frac{8700779}{3189540}a^{7}+\frac{22770689}{2551632}a^{6}-\frac{38283971}{2126360}a^{5}+\frac{15375181}{375240}a^{4}-\frac{232852769}{4252720}a^{3}+\frac{127971217}{1594770}a^{2}-\frac{556353}{10030}a+\frac{55831}{2006}$, $\frac{1}{212636}a^{14}-\frac{7}{212636}a^{13}+\frac{7561}{12758160}a^{12}-\frac{6651}{2126360}a^{11}+\frac{11799}{425272}a^{10}-\frac{282811}{2551632}a^{9}+\frac{982109}{1594770}a^{8}-\frac{5841241}{3189540}a^{7}+\frac{293053}{43248}a^{6}-\frac{91546307}{6379080}a^{5}+\frac{222034363}{6379080}a^{4}-\frac{607624493}{12758160}a^{3}+\frac{219252727}{3189540}a^{2}-\frac{94373}{2006}a+\frac{740131}{30090}$, $\frac{29}{2126360}a^{14}-\frac{203}{2126360}a^{13}+\frac{15377}{12758160}a^{12}-\frac{6369}{1063180}a^{11}+\frac{25423}{797385}a^{10}-\frac{454093}{4252720}a^{9}+\frac{8039}{40120}a^{8}-\frac{473813}{2126360}a^{7}-\frac{2275827}{850544}a^{6}+\frac{8952431}{1063180}a^{5}-\frac{27937318}{797385}a^{4}+\frac{713959459}{12758160}a^{3}-\frac{16374994}{159477}a^{2}+\frac{2290211}{30090}a-\frac{403859}{10030}$, $\frac{1}{37524}a^{14}-\frac{7}{37524}a^{13}+\frac{3279}{1063180}a^{12}-\frac{51287}{3189540}a^{11}+\frac{83183}{637908}a^{10}-\frac{324725}{637908}a^{9}+\frac{683602}{265795}a^{8}-\frac{7870331}{1063180}a^{7}+\frac{15705083}{637908}a^{6}-\frac{159218363}{3189540}a^{5}+\frac{337374217}{3189540}a^{4}-\frac{25526623}{187620}a^{3}+\frac{165336207}{1063180}a^{2}-\frac{284981}{3009}a-\frac{40756}{15045}$, $\frac{149}{6379080}a^{14}-\frac{1043}{6379080}a^{13}+\frac{33827}{12758160}a^{12}-\frac{43961}{3189540}a^{11}+\frac{29182}{265795}a^{10}-\frac{1813831}{4252720}a^{9}+\frac{907553}{425272}a^{8}-\frac{13017743}{2126360}a^{7}+\frac{52789685}{2551632}a^{6}-\frac{7933561}{187620}a^{5}+\frac{26051824}{265795}a^{4}-\frac{1683001351}{12758160}a^{3}+\frac{325800823}{1594770}a^{2}-\frac{85243}{590}a+\frac{1356673}{10030}$ 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:  \( 6878292.5579 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 6878292.5579 \cdot 484}{2\cdot\sqrt{10483151353726139536553735554369}}\cr\approx \mathstrut & 1.2487888970 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008)
 
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 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008, 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 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008);
 
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 - 8*x^15 + 132*x^14 - 784*x^13 + 6578*x^12 - 29640*x^11 + 161878*x^10 - 555422*x^9 + 2130764*x^8 - 5486248*x^7 + 15012390*x^6 - 27873848*x^5 + 52839433*x^4 - 64659186*x^3 + 76467296*x^2 - 48013336*x + 22193008);
 
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

$\SD_{16}$ (as 16T12):

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 16
The 7 conjugacy class representatives for $QD_{16}$
Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{901}) \), \(\Q(\sqrt{17}, \sqrt{53})\), 4.4.260389.1 x2, 4.4.13800617.1 x2, 8.8.190457029580689.1, 8.0.61089990620221.2 x4

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 8 sibling: 8.0.61089990620221.2
Minimal sibling: 8.0.61089990620221.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.2.0.1}{2} }^{8}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.8.0.1}{8} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ R ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(17\) Copy content Toggle raw display 17.8.7.4$x^{8} + 34$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.4$x^{8} + 34$$8$$1$$7$$C_8$$[\ ]_{8}$
\(53\) Copy content Toggle raw display 53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} + 53$$2$$1$$1$$C_2$$[\ ]_{2}$