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Properties

Label 16.0.22847714566...5201.3
Degree $16$
Signature $[0, 8]$
Discriminant $29^{12}\cdot 71^{8}$
Root discriminant $105.30$
Ramified primes $29, 71$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30084761, -103177579, 154253711, -130400988, 64709569, -15542808, -1227572, 2056869, -604110, 39299, 33304, -11388, 361, 440, -57, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 57*x^14 + 440*x^13 + 361*x^12 - 11388*x^11 + 33304*x^10 + 39299*x^9 - 604110*x^8 + 2056869*x^7 - 1227572*x^6 - 15542808*x^5 + 64709569*x^4 - 130400988*x^3 + 154253711*x^2 - 103177579*x + 30084761)
 
gp: K = bnfinit(x^16 - 5*x^15 - 57*x^14 + 440*x^13 + 361*x^12 - 11388*x^11 + 33304*x^10 + 39299*x^9 - 604110*x^8 + 2056869*x^7 - 1227572*x^6 - 15542808*x^5 + 64709569*x^4 - 130400988*x^3 + 154253711*x^2 - 103177579*x + 30084761, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 57 x^{14} + 440 x^{13} + 361 x^{12} - 11388 x^{11} + 33304 x^{10} + 39299 x^{9} - 604110 x^{8} + 2056869 x^{7} - 1227572 x^{6} - 15542808 x^{5} + 64709569 x^{4} - 130400988 x^{3} + 154253711 x^{2} - 103177579 x + 30084761 \)

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:  \(228477145661885006672293047985201=29^{12}\cdot 71^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.30$
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:  $29, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{52} a^{13} + \frac{1}{26} a^{12} + \frac{5}{52} a^{11} + \frac{2}{13} a^{10} - \frac{7}{26} a^{9} + \frac{19}{52} a^{8} + \frac{3}{13} a^{7} - \frac{1}{52} a^{6} + \frac{3}{13} a^{5} + \frac{5}{26} a^{4} - \frac{21}{52} a^{3} - \frac{11}{26} a^{2} - \frac{17}{52} a - \frac{1}{26}$, $\frac{1}{4316} a^{14} + \frac{33}{4316} a^{13} - \frac{58}{1079} a^{12} + \frac{1073}{4316} a^{11} + \frac{25}{332} a^{10} - \frac{2001}{4316} a^{9} - \frac{361}{4316} a^{8} - \frac{367}{2158} a^{7} + \frac{1229}{4316} a^{6} + \frac{1279}{4316} a^{5} - \frac{829}{4316} a^{4} - \frac{387}{4316} a^{3} + \frac{186}{1079} a^{2} - \frac{555}{4316} a - \frac{15}{52}$, $\frac{1}{97708352437438056347839321698931005987021656} a^{15} - \frac{2269302052758230983613221246891608363511}{48854176218719028173919660849465502993510828} a^{14} + \frac{386049597876897965510530849527742749774707}{97708352437438056347839321698931005987021656} a^{13} + \frac{582321961966748587489813911039116703117851}{7516027110572158180603024746071615845155512} a^{12} - \frac{395504489533625541123334065257944853505081}{1879006777643039545150756186517903961288878} a^{11} + \frac{14104453988345283299702378186368595679813651}{48854176218719028173919660849465502993510828} a^{10} - \frac{2344044336941129994302651374505960063823395}{6979168031245575453417094407066500427644404} a^{9} + \frac{5196437874629002337967659111067990491222843}{97708352437438056347839321698931005987021656} a^{8} + \frac{15540042861701326640802732071797582077977969}{97708352437438056347839321698931005987021656} a^{7} - \frac{1025914900016352066638100661988244757891199}{6979168031245575453417094407066500427644404} a^{6} - \frac{5711225891695469277712593245602534792997932}{12213544054679757043479915212366375748377707} a^{5} + \frac{8040661174342062368957817031870049880877243}{24427088109359514086959830424732751496755414} a^{4} + \frac{42775866741946599558945346075723477630915891}{97708352437438056347839321698931005987021656} a^{3} + \frac{16997658558483946512242539932375890368375719}{97708352437438056347839321698931005987021656} a^{2} - \frac{7352381645320858550658552255120084812292833}{48854176218719028173919660849465502993510828} a + \frac{34434859226375000109978091680030015255193}{168172723644471697672701070049795191027576}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 488606731.59 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.2.59711.1, 4.0.24389.1, 4.2.1731619.1, 8.4.521222775331469.1, 8.0.521222775331469.1, 8.0.2998504361161.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.8.6.1$x^{8} - 14129 x^{4} + 73805281$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$