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Properties

Label 16.8.27248404294...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{6}\cdot 29^{12}\cdot 149^{4}$
Root discriminant $79.84$
Ramified primes $5, 29, 149$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T382)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-166291, 68512, 87664, -362051, 310144, 67167, -153047, 50772, 1739, -12099, 6443, -285, -532, 145, -7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 7*x^14 + 145*x^13 - 532*x^12 - 285*x^11 + 6443*x^10 - 12099*x^9 + 1739*x^8 + 50772*x^7 - 153047*x^6 + 67167*x^5 + 310144*x^4 - 362051*x^3 + 87664*x^2 + 68512*x - 166291)
 
gp: K = bnfinit(x^16 - 6*x^15 - 7*x^14 + 145*x^13 - 532*x^12 - 285*x^11 + 6443*x^10 - 12099*x^9 + 1739*x^8 + 50772*x^7 - 153047*x^6 + 67167*x^5 + 310144*x^4 - 362051*x^3 + 87664*x^2 + 68512*x - 166291, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 7 x^{14} + 145 x^{13} - 532 x^{12} - 285 x^{11} + 6443 x^{10} - 12099 x^{9} + 1739 x^{8} + 50772 x^{7} - 153047 x^{6} + 67167 x^{5} + 310144 x^{4} - 362051 x^{3} + 87664 x^{2} + 68512 x - 166291 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2724840429455819815583272515625=5^{6}\cdot 29^{12}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.84$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{38342824258692754364069309461888043123775787} a^{15} + \frac{2620671654880097846188768888157595429151310}{5477546322670393480581329923126863303396541} a^{14} + \frac{769164723879153369806235150783706553500170}{5477546322670393480581329923126863303396541} a^{13} + \frac{19137001655940096502821233937435292053438967}{38342824258692754364069309461888043123775787} a^{12} - \frac{875442105216081455540922397396205930611998}{38342824258692754364069309461888043123775787} a^{11} - \frac{2157016149266609282237387410363663175329294}{5477546322670393480581329923126863303396541} a^{10} - \frac{14864309875560539971894471277116362146659739}{38342824258692754364069309461888043123775787} a^{9} - \frac{2685713439338167424863818633599629753033004}{38342824258692754364069309461888043123775787} a^{8} + \frac{10077969246420919165833549189288902956734265}{38342824258692754364069309461888043123775787} a^{7} - \frac{6747675545561906222139741076746397161663544}{38342824258692754364069309461888043123775787} a^{6} + \frac{9550478442267045066738713927003838071111982}{38342824258692754364069309461888043123775787} a^{5} + \frac{1489706677175872706027019530349397767679597}{5477546322670393480581329923126863303396541} a^{4} - \frac{1580233062368794718894511143604097941034801}{38342824258692754364069309461888043123775787} a^{3} + \frac{14355378238065961073203581852005331734188754}{38342824258692754364069309461888043123775787} a^{2} + \frac{19031175109114074794307608823906746006149172}{38342824258692754364069309461888043123775787} a - \frac{64159152570071783928236968144372630843797}{214205722115601979687538041686525380579753}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 3086199522.64 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T382):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.3633961.1, 4.4.4205.1, 4.4.18169805.2, 8.8.330141813738025.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29Data not computed
$149$149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$