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Properties

Label 16.8.27583540507...0625.2
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 29^{8}\cdot 109^{4}$
Root discriminant $38.91$
Ramified primes $5, 29, 109$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2.C_2\wr C_2^2$ (as 16T394)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-49, 6139, 17533, -23682, -30095, 42105, -32927, 27373, -10015, 4382, -1180, -24, -16, -52, 11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 11*x^14 - 52*x^13 - 16*x^12 - 24*x^11 - 1180*x^10 + 4382*x^9 - 10015*x^8 + 27373*x^7 - 32927*x^6 + 42105*x^5 - 30095*x^4 - 23682*x^3 + 17533*x^2 + 6139*x - 49)
 
gp: K = bnfinit(x^16 - 3*x^15 + 11*x^14 - 52*x^13 - 16*x^12 - 24*x^11 - 1180*x^10 + 4382*x^9 - 10015*x^8 + 27373*x^7 - 32927*x^6 + 42105*x^5 - 30095*x^4 - 23682*x^3 + 17533*x^2 + 6139*x - 49, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 11 x^{14} - 52 x^{13} - 16 x^{12} - 24 x^{11} - 1180 x^{10} + 4382 x^{9} - 10015 x^{8} + 27373 x^{7} - 32927 x^{6} + 42105 x^{5} - 30095 x^{4} - 23682 x^{3} + 17533 x^{2} + 6139 x - 49 \)

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:  \(27583540507977079969140625=5^{8}\cdot 29^{8}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.91$
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:  $5, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{35} a^{14} - \frac{1}{7} a^{13} - \frac{17}{35} a^{11} - \frac{17}{35} a^{10} + \frac{17}{35} a^{9} - \frac{17}{35} a^{8} - \frac{1}{35} a^{7} + \frac{4}{35} a^{6} + \frac{16}{35} a^{5} - \frac{3}{35} a^{4} - \frac{3}{7} a^{3} - \frac{1}{5} a^{2} - \frac{8}{35} a - \frac{2}{5}$, $\frac{1}{291656940628781903354195019869075} a^{15} - \frac{3603164052588948998234373317454}{291656940628781903354195019869075} a^{14} + \frac{8987288968021358571707843562186}{58331388125756380670839003973815} a^{13} - \frac{27683492626173646136017886797057}{291656940628781903354195019869075} a^{12} - \frac{9893231942267086009867148557787}{41665277232683129050599288552725} a^{11} - \frac{21377983571911605777821231139734}{58331388125756380670839003973815} a^{10} - \frac{20641715186663779364885156858868}{58331388125756380670839003973815} a^{9} + \frac{68285800100389309684286920327352}{291656940628781903354195019869075} a^{8} - \frac{116609207817502253691419602024972}{291656940628781903354195019869075} a^{7} - \frac{23567680006943566289075677387969}{58331388125756380670839003973815} a^{6} + \frac{51558620564978049457680874361403}{291656940628781903354195019869075} a^{5} - \frac{43827457703290651064161661907108}{291656940628781903354195019869075} a^{4} - \frac{128748298257528874237074931155132}{291656940628781903354195019869075} a^{3} - \frac{2272705276645858549096633883826}{11666277625151276134167800794763} a^{2} - \frac{35304336165052099155847031674772}{291656940628781903354195019869075} a + \frac{19127982962063323303362523998063}{41665277232683129050599288552725}$

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:  \( 4662800.92344 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2.C_2\wr C_2^2$ (as 16T394):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 17 conjugacy class representatives for $C_2.C_2\wr C_2^2$
Character table for $C_2.C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.4.4205.1 x2, 4.4.725.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
109Data not computed