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Properties

Label 16.0.80073733711...8449.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 73^{14}$
Root discriminant $73.96$
Ramified primes $3, 73$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![262144, 188416, -149248, 749376, 581200, -100624, 275740, -118276, 123893, -55975, 23233, -6102, 1434, -250, 37, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 37*x^14 - 250*x^13 + 1434*x^12 - 6102*x^11 + 23233*x^10 - 55975*x^9 + 123893*x^8 - 118276*x^7 + 275740*x^6 - 100624*x^5 + 581200*x^4 + 749376*x^3 - 149248*x^2 + 188416*x + 262144)
 
gp: K = bnfinit(x^16 - 3*x^15 + 37*x^14 - 250*x^13 + 1434*x^12 - 6102*x^11 + 23233*x^10 - 55975*x^9 + 123893*x^8 - 118276*x^7 + 275740*x^6 - 100624*x^5 + 581200*x^4 + 749376*x^3 - 149248*x^2 + 188416*x + 262144, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 37 x^{14} - 250 x^{13} + 1434 x^{12} - 6102 x^{11} + 23233 x^{10} - 55975 x^{9} + 123893 x^{8} - 118276 x^{7} + 275740 x^{6} - 100624 x^{5} + 581200 x^{4} + 749376 x^{3} - 149248 x^{2} + 188416 x + 262144 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(800737337114777368288115578449=3^{8}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.96$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{192} a^{12} - \frac{5}{192} a^{11} + \frac{1}{64} a^{10} - \frac{1}{48} a^{9} - \frac{3}{32} a^{8} - \frac{1}{96} a^{7} + \frac{15}{64} a^{6} + \frac{7}{192} a^{5} - \frac{37}{192} a^{4} + \frac{17}{96} a^{3} + \frac{7}{16} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{768} a^{13} - \frac{1}{768} a^{12} - \frac{17}{768} a^{11} + \frac{7}{96} a^{10} - \frac{17}{384} a^{9} - \frac{13}{384} a^{8} - \frac{11}{768} a^{7} - \frac{53}{768} a^{6} + \frac{45}{256} a^{5} - \frac{3}{128} a^{4} - \frac{5}{192} a^{3} - \frac{1}{16} a^{2} - \frac{5}{12} a + \frac{1}{3}$, $\frac{1}{102833664} a^{14} - \frac{51353}{102833664} a^{13} + \frac{154087}{102833664} a^{12} + \frac{63079}{2142368} a^{11} + \frac{1790565}{17138944} a^{10} - \frac{1082807}{17138944} a^{9} - \frac{3598665}{34277888} a^{8} - \frac{8266285}{102833664} a^{7} - \frac{11672929}{102833664} a^{6} + \frac{9047011}{51416832} a^{5} + \frac{5557367}{25708416} a^{4} - \frac{357119}{6427104} a^{3} + \frac{112799}{803388} a^{2} + \frac{18365}{267796} a + \frac{16382}{200847}$, $\frac{1}{257920843612230179639242404421632} a^{15} + \frac{126626448934918306001167}{85973614537410059879747468140544} a^{14} + \frac{37551170710823427548948435029}{257920843612230179639242404421632} a^{13} - \frac{154608317233388349972900242309}{128960421806115089819621202210816} a^{12} - \frac{423338691874953983654662874723}{128960421806115089819621202210816} a^{11} - \frac{9506907235539702887480540458331}{128960421806115089819621202210816} a^{10} - \frac{22007536082076995069750897553119}{257920843612230179639242404421632} a^{9} - \frac{5507039651258892717436265390461}{85973614537410059879747468140544} a^{8} - \frac{20859601761759786571347732538651}{257920843612230179639242404421632} a^{7} - \frac{8288061278081489895582160944197}{64480210903057544909810601105408} a^{6} + \frac{8841524905954804512947619879031}{64480210903057544909810601105408} a^{5} - \frac{2334095001086766145666850203985}{16120052725764386227452650276352} a^{4} + \frac{4583012220370273925455279376429}{16120052725764386227452650276352} a^{3} - \frac{236135595721325880961803983731}{4030013181441096556863162569088} a^{2} - \frac{174555886077353655838348567159}{1007503295360274139215790642272} a + \frac{8220040866245521852199863679}{31484477980008566850493457571}$

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

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{18439819587700349913463}{3852497327999375340023635968} a^{15} - \frac{74825824784162538623749}{3852497327999375340023635968} a^{14} + \frac{250117909405872326867505}{1284165775999791780007878656} a^{13} - \frac{2684332261082741270064235}{1926248663999687670011817984} a^{12} + \frac{15855785331725955672177067}{1926248663999687670011817984} a^{11} - \frac{71614528428560019592933853}{1926248663999687670011817984} a^{10} + \frac{563718753153963304419729271}{3852497327999375340023635968} a^{9} - \frac{519640758846293059440619883}{1284165775999791780007878656} a^{8} + \frac{1222739099617951494456475201}{1284165775999791780007878656} a^{7} - \frac{1351242251808064830702549919}{963124331999843835005908992} a^{6} + \frac{2334550280923923409716149057}{963124331999843835005908992} a^{5} - \frac{629389218656614501509898783}{240781082999960958751477248} a^{4} + \frac{1107296633680051194307328755}{240781082999960958751477248} a^{3} - \frac{25952723125308525505563629}{60195270749990239687869312} a^{2} - \frac{29490615235047830208734881}{15048817687497559921967328} a + \frac{378322088213908453639368}{156758517578099582520493} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 774296732.554 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-219}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{73})\), 4.4.389017.1, 4.0.3501153.1, 8.0.12258072329409.1, 8.4.99426586671873.1 x2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$73$73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$