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Properties

Label 16.0.43538996073...0625.9
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 61^{10}$
Root discriminant $53.39$
Ramified primes $5, 61$
Class number $200$ (GRH)
Class group $[2, 10, 10]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![283282561, 0, 41074633, 0, 48780893, 0, -23532134, 0, 3505530, 0, -242414, 0, 8533, 0, -147, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 147*x^14 + 8533*x^12 - 242414*x^10 + 3505530*x^8 - 23532134*x^6 + 48780893*x^4 + 41074633*x^2 + 283282561)
 
gp: K = bnfinit(x^16 - 147*x^14 + 8533*x^12 - 242414*x^10 + 3505530*x^8 - 23532134*x^6 + 48780893*x^4 + 41074633*x^2 + 283282561, 1)
 

Normalized defining polynomial

\( x^{16} - 147 x^{14} + 8533 x^{12} - 242414 x^{10} + 3505530 x^{8} - 23532134 x^{6} + 48780893 x^{4} + 41074633 x^{2} + 283282561 \)

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:  \(4353899607317398687744140625=5^{14}\cdot 61^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.39$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{2} + \frac{3}{8} a + \frac{1}{8}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{6} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{3}{16} a - \frac{1}{16}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a + \frac{3}{16}$, $\frac{1}{64} a^{12} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{32} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{16} a^{3} + \frac{1}{16} a^{2} + \frac{5}{16} a + \frac{1}{64}$, $\frac{1}{64} a^{13} - \frac{1}{16} a^{8} + \frac{1}{32} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{4} - \frac{7}{16} a^{2} - \frac{19}{64} a - \frac{5}{16}$, $\frac{1}{6245808229593237454080128} a^{14} - \frac{1}{128} a^{13} + \frac{31732129494206306927351}{6245808229593237454080128} a^{12} - \frac{1}{32} a^{11} - \frac{7875093420244610530239}{390363014349577340880008} a^{10} - \frac{1}{32} a^{9} - \frac{117945912242894091913917}{3122904114796618727040064} a^{8} + \frac{3}{64} a^{7} - \frac{77460688593705156126987}{3122904114796618727040064} a^{6} + \frac{1}{32} a^{5} - \frac{6055468891525969999755}{48795376793697167610001} a^{4} - \frac{3}{32} a^{3} + \frac{1889134540226961203169161}{6245808229593237454080128} a^{2} + \frac{27}{128} a - \frac{363963020660540726175681}{6245808229593237454080128}$, $\frac{1}{105123198312283779589622634368} a^{15} - \frac{75427586584769833190744377}{13140399789035472448702829296} a^{13} - \frac{1}{128} a^{12} + \frac{205933785072514766039693265}{26280799578070944897405658592} a^{11} - \frac{1}{32} a^{10} - \frac{2116958981976413419348977299}{52561599156141889794811317184} a^{9} - \frac{1}{32} a^{8} - \frac{436342591945140076762887435}{26280799578070944897405658592} a^{7} - \frac{5}{64} a^{6} - \frac{267055690689258640699087629}{26280799578070944897405658592} a^{5} + \frac{1}{32} a^{4} - \frac{19107356325407457818225422459}{105123198312283779589622634368} a^{3} + \frac{13}{32} a^{2} + \frac{3887595263373267724101968253}{26280799578070944897405658592} a + \frac{43}{128}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}$, which has order $200$ (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:  \( \frac{2366466318975711877}{129621699521928211577833088} a^{15} + \frac{10433131}{32197425071276944} a^{14} - \frac{165287015537137563035}{64810849760964105788916544} a^{13} + \frac{1029554417}{257579400570215552} a^{12} + \frac{4447264343464154255441}{32405424880482052894458272} a^{11} - \frac{253209079887}{64394850142553888} a^{10} - \frac{222813125146135379144047}{64810849760964105788916544} a^{9} + \frac{15520150859219}{64394850142553888} a^{8} + \frac{645214003735840854990551}{16202712440241026447229136} a^{7} - \frac{673750334499783}{128789700285107776} a^{6} - \frac{5365048585397481453149277}{32405424880482052894458272} a^{5} + \frac{2326223540962391}{64394850142553888} a^{4} - \frac{244284142220286739849095}{129621699521928211577833088} a^{3} + \frac{3546866266408939}{64394850142553888} a^{2} + \frac{838473051773102906189583}{64810849760964105788916544} a - \frac{40819238146423459}{257579400570215552} \) (order $10$)
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:  \( 503142.791513 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.7625.1, \(\Q(\zeta_{5})\), 4.0.1525.1, 8.8.65984086015625.1, 8.0.65984086015625.1, 8.0.58140625.2

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.2.2$x^{4} - 61 x^{2} + 7442$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
61.4.3.4$x^{4} + 488$$4$$1$$3$$C_4$$[\ ]_{4}$