↑

Properties

Label 16.0.21652340025...441.10
Degree $16$
Signature $[0, 8]$
Discriminant $13^{8}\cdot 61^{12}$
Root discriminant $78.70$
Ramified primes $13, 61$
Class number $576$ (GRH)
Class group $[4, 144]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![171239, -389678, 787813, -873606, 840967, -583197, 381852, -171376, 80234, -26156, 9153, -2213, 689, -128, 33, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 33*x^14 - 128*x^13 + 689*x^12 - 2213*x^11 + 9153*x^10 - 26156*x^9 + 80234*x^8 - 171376*x^7 + 381852*x^6 - 583197*x^5 + 840967*x^4 - 873606*x^3 + 787813*x^2 - 389678*x + 171239)
 
gp: K = bnfinit(x^16 - 2*x^15 + 33*x^14 - 128*x^13 + 689*x^12 - 2213*x^11 + 9153*x^10 - 26156*x^9 + 80234*x^8 - 171376*x^7 + 381852*x^6 - 583197*x^5 + 840967*x^4 - 873606*x^3 + 787813*x^2 - 389678*x + 171239, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 33 x^{14} - 128 x^{13} + 689 x^{12} - 2213 x^{11} + 9153 x^{10} - 26156 x^{9} + 80234 x^{8} - 171376 x^{7} + 381852 x^{6} - 583197 x^{5} + 840967 x^{4} - 873606 x^{3} + 787813 x^{2} - 389678 x + 171239 \)

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:  \(2165234002589380425486809479441=13^{8}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.70$
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:  $13, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{15} a^{12} + \frac{1}{15} a^{10} - \frac{1}{3} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{4}{15} a^{6} - \frac{2}{5} a^{5} - \frac{1}{3} a^{4} - \frac{1}{5} a^{3} + \frac{4}{15} a + \frac{4}{15}$, $\frac{1}{15} a^{13} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{3} a^{7} + \frac{1}{5} a^{6} + \frac{4}{15} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{3} a^{2} + \frac{4}{15} a + \frac{2}{5}$, $\frac{1}{15} a^{14} + \frac{1}{15} a^{11} - \frac{1}{15} a^{10} - \frac{4}{15} a^{9} - \frac{1}{15} a^{8} - \frac{4}{15} a^{6} + \frac{1}{3} a^{4} + \frac{7}{15} a^{3} + \frac{1}{15} a^{2} + \frac{2}{15} a - \frac{7}{15}$, $\frac{1}{81573724579864402540760615718933806925} a^{15} + \frac{72745774742433347443179770397491089}{81573724579864402540760615718933806925} a^{14} - \frac{326920563893768479723481534189583158}{81573724579864402540760615718933806925} a^{13} + \frac{221634237504586054829664527784076004}{81573724579864402540760615718933806925} a^{12} + \frac{6936670195177224480247003123921608193}{81573724579864402540760615718933806925} a^{11} + \frac{29935821710498794821163943384360298}{5438248305324293502717374381262253795} a^{10} - \frac{19933290983900217869625795263529667627}{81573724579864402540760615718933806925} a^{9} + \frac{9028157144354068108578238658920562}{81573724579864402540760615718933806925} a^{8} - \frac{10253208496355080722717145258756700459}{81573724579864402540760615718933806925} a^{7} - \frac{1585357815660869200855841637276697183}{16314744915972880508152123143786761385} a^{6} + \frac{5204659595650924727531672708095643494}{27191241526621467513586871906311268975} a^{5} - \frac{7444194362475230677987886681271890704}{16314744915972880508152123143786761385} a^{4} - \frac{4753126317331483249482353274207923278}{81573724579864402540760615718933806925} a^{3} + \frac{36671891266137393214858408141927006111}{81573724579864402540760615718933806925} a^{2} - \frac{729280258628028711199614952734205897}{27191241526621467513586871906311268975} a + \frac{8236879102373172681765179628643706962}{27191241526621467513586871906311268975}$

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

Class group and class number

$C_{4}\times C_{144}$, which has order $576$ (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:  \( 996723.055385 \) (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

\(\Q(\sqrt{61}) \), 4.4.48373.1, 4.0.38359789.2, 4.0.2950753.1, 8.4.142736774869.1, 8.4.24122514952861.3, 8.0.1471473412124521.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$61$61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61.8.6.1$x^{8} - 61 x^{4} + 59536$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$