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Properties

Label 16.0.22034658496...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 41^{12}$
Root discriminant $44.30$
Ramified primes $5, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52645, 66550, 98626, -41412, -121567, 10856, 45751, -7008, -6440, 2168, 255, -438, 141, 36, -17, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 17*x^14 + 36*x^13 + 141*x^12 - 438*x^11 + 255*x^10 + 2168*x^9 - 6440*x^8 - 7008*x^7 + 45751*x^6 + 10856*x^5 - 121567*x^4 - 41412*x^3 + 98626*x^2 + 66550*x + 52645)
 
gp: K = bnfinit(x^16 - 2*x^15 - 17*x^14 + 36*x^13 + 141*x^12 - 438*x^11 + 255*x^10 + 2168*x^9 - 6440*x^8 - 7008*x^7 + 45751*x^6 + 10856*x^5 - 121567*x^4 - 41412*x^3 + 98626*x^2 + 66550*x + 52645, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 17 x^{14} + 36 x^{13} + 141 x^{12} - 438 x^{11} + 255 x^{10} + 2168 x^{9} - 6440 x^{8} - 7008 x^{7} + 45751 x^{6} + 10856 x^{5} - 121567 x^{4} - 41412 x^{3} + 98626 x^{2} + 66550 x + 52645 \)

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:  \(220346584964513535947265625=5^{10}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.30$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{80} a^{14} + \frac{9}{80} a^{13} + \frac{3}{80} a^{12} - \frac{1}{40} a^{11} + \frac{1}{40} a^{10} - \frac{9}{40} a^{9} + \frac{19}{80} a^{8} - \frac{1}{80} a^{7} + \frac{1}{10} a^{6} - \frac{1}{80} a^{5} + \frac{7}{20} a^{4} + \frac{3}{80} a^{3} + \frac{7}{40} a^{2} + \frac{5}{16} a + \frac{7}{16}$, $\frac{1}{90131646206749786940896668901723600} a^{15} + \frac{77538182061724292875157506960189}{22532911551687446735224167225430900} a^{14} - \frac{4450621635353298259100703637530547}{45065823103374893470448334450861800} a^{13} + \frac{10382184118351621897509810921976659}{90131646206749786940896668901723600} a^{12} + \frac{3684292629558398115124494758077747}{22532911551687446735224167225430900} a^{11} + \frac{5004753060228108348379346629674829}{22532911551687446735224167225430900} a^{10} + \frac{1266039644811098289855185203215933}{90131646206749786940896668901723600} a^{9} + \frac{682847928965375624154553552412879}{11266455775843723367612083612715450} a^{8} + \frac{14386124719844100851310451256699241}{90131646206749786940896668901723600} a^{7} - \frac{923453904992828490375317909957401}{18026329241349957388179333780344720} a^{6} + \frac{31393922910339367750653385606724261}{90131646206749786940896668901723600} a^{5} + \frac{7860554570303711132154680286970819}{90131646206749786940896668901723600} a^{4} - \frac{769944299283515228073800957565193}{18026329241349957388179333780344720} a^{3} - \frac{482133359107232402763278574321123}{1527655020453386219337231676300400} a^{2} + \frac{2563243174213811065445662817149797}{9013164620674978694089666890172360} a - \frac{441148979497346207917311830655343}{18026329241349957388179333780344720}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 12961354.7506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{205}) \), 4.0.8405.1 x2, 4.0.1025.1 x2, \(\Q(\sqrt{5}, \sqrt{41})\), 8.4.14844075753125.1 x2, 8.0.2968815150625.1 x2, 8.0.1766100625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$41$41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41.8.6.1$x^{8} - 9881 x^{4} + 34857216$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$