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Properties

Label 16.0.76304811594...169.12
Degree $16$
Signature $[0, 8]$
Discriminant $43^{8}\cdot 97^{14}$
Root discriminant $359.05$
Ramified primes $43, 97$
Class number $43520$ (GRH)
Class group $[4, 4, 2720]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23997286772, -58470158494, 58389670393, -31677700420, 10962296202, -2968080528, 764759651, -186139766, 37570315, -5247324, 434923, 7562, -5921, 686, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 + 686*x^13 - 5921*x^12 + 7562*x^11 + 434923*x^10 - 5247324*x^9 + 37570315*x^8 - 186139766*x^7 + 764759651*x^6 - 2968080528*x^5 + 10962296202*x^4 - 31677700420*x^3 + 58389670393*x^2 - 58470158494*x + 23997286772)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 + 686*x^13 - 5921*x^12 + 7562*x^11 + 434923*x^10 - 5247324*x^9 + 37570315*x^8 - 186139766*x^7 + 764759651*x^6 - 2968080528*x^5 + 10962296202*x^4 - 31677700420*x^3 + 58389670393*x^2 - 58470158494*x + 23997286772, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} + 686 x^{13} - 5921 x^{12} + 7562 x^{11} + 434923 x^{10} - 5247324 x^{9} + 37570315 x^{8} - 186139766 x^{7} + 764759651 x^{6} - 2968080528 x^{5} + 10962296202 x^{4} - 31677700420 x^{3} + 58389670393 x^{2} - 58470158494 x + 23997286772 \)

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:  \(76304811594430773788792663582199493515169=43^{8}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $359.05$
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:  $43, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{6} - \frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{24} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{6}$, $\frac{1}{24} a^{9} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{6} a^{4} + \frac{3}{8} a^{3} - \frac{1}{12} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{48} a^{10} - \frac{1}{48} a^{9} - \frac{1}{48} a^{8} - \frac{1}{24} a^{6} - \frac{1}{8} a^{5} - \frac{7}{48} a^{4} + \frac{5}{16} a^{3} - \frac{3}{16} a^{2} + \frac{11}{24} a - \frac{1}{4}$, $\frac{1}{14832} a^{11} + \frac{1}{144} a^{10} - \frac{241}{14832} a^{9} + \frac{79}{7416} a^{8} - \frac{143}{7416} a^{7} + \frac{7}{7416} a^{6} - \frac{1607}{14832} a^{5} - \frac{2393}{14832} a^{4} - \frac{7397}{14832} a^{3} + \frac{179}{1854} a^{2} - \frac{343}{927} a + \frac{265}{1854}$, $\frac{1}{2551104} a^{12} + \frac{5}{318888} a^{11} - \frac{3949}{2551104} a^{10} - \frac{18131}{1275552} a^{9} - \frac{3751}{2551104} a^{8} + \frac{13069}{318888} a^{7} + \frac{188473}{2551104} a^{6} + \frac{31193}{1275552} a^{5} - \frac{258029}{2551104} a^{4} + \frac{60323}{318888} a^{3} + \frac{537437}{2551104} a^{2} - \frac{606911}{1275552} a - \frac{12671}{70864}$, $\frac{1}{22959936} a^{13} + \frac{1}{11479968} a^{12} - \frac{481}{22959936} a^{11} + \frac{2669}{637776} a^{10} + \frac{278393}{22959936} a^{9} - \frac{120179}{11479968} a^{8} - \frac{108863}{22959936} a^{7} + \frac{34247}{1913328} a^{6} - \frac{439997}{22959936} a^{5} + \frac{716993}{11479968} a^{4} + \frac{10037273}{22959936} a^{3} + \frac{811823}{1913328} a^{2} + \frac{41393}{106296} a - \frac{231799}{2869992}$, $\frac{1}{79165859328} a^{14} + \frac{1085}{79165859328} a^{13} - \frac{1073}{9895732416} a^{12} - \frac{21565}{26388619776} a^{11} + \frac{350594881}{39582929664} a^{10} - \frac{706549129}{79165859328} a^{9} + \frac{796361111}{39582929664} a^{8} + \frac{57234511}{8796206592} a^{7} + \frac{1091531945}{39582929664} a^{6} + \frac{16437183625}{79165859328} a^{5} + \frac{1969708403}{19791464832} a^{4} - \frac{3410304215}{26388619776} a^{3} + \frac{930097853}{2932068864} a^{2} + \frac{5034771407}{39582929664} a - \frac{1528467857}{6597154944}$, $\frac{1}{66718204745448418004517559137453528576} a^{15} - \frac{402028774509434475585890437}{66718204745448418004517559137453528576} a^{14} + \frac{347373286293836732915840689823}{33359102372724209002258779568726764288} a^{13} - \frac{9100350473949716082234273040175}{66718204745448418004517559137453528576} a^{12} - \frac{28105763190575447317380786077087}{1389962598863508708427449148696948512} a^{11} + \frac{469729273844108900101888614087440699}{66718204745448418004517559137453528576} a^{10} - \frac{31399692797855041197213230396095135}{4169887796590526125282347446090845536} a^{9} + \frac{1326062975268264293643930163299102979}{66718204745448418004517559137453528576} a^{8} + \frac{43616232693822915094666860078112297}{1853283465151344944569932198262598016} a^{7} + \frac{1797751696058339987609832540950836957}{66718204745448418004517559137453528576} a^{6} + \frac{1367712526777047878270233396811665589}{33359102372724209002258779568726764288} a^{5} + \frac{11960589831152412031202002522411108139}{66718204745448418004517559137453528576} a^{4} - \frac{28234065738238308175072194848921152295}{66718204745448418004517559137453528576} a^{3} - \frac{1893509095534632052401788361434168881}{8339775593181052250564694892181691072} a^{2} - \frac{4347095828397520238322986759664451}{926641732575672472284966099131299008} a - \frac{3833724987871051925238859571486830913}{8339775593181052250564694892181691072}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{2720}$, which has order $43520$ (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:  \( -1 \) (order $2$)
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:  \( 29102662061.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{97}) \), \(\Q(\sqrt{-4171}) \), \(\Q(\sqrt{-43}) \), 4.4.912673.1, 4.0.1687532377.2, \(\Q(\sqrt{-43}, \sqrt{97})\), 8.4.149396028000030937.3 x2, 8.0.276233255772057202513.1 x2, 8.0.2847765523423270129.2

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 siblings: 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$