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Properties

Label 16.8.15712755557...5489.6
Degree $16$
Signature $[8, 4]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6534883, 8298938, -2370356, -6299140, -1607860, 1147826, 1107150, 106202, -262349, -48182, 27164, 4002, -819, -64, -31, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 31*x^14 - 64*x^13 - 819*x^12 + 4002*x^11 + 27164*x^10 - 48182*x^9 - 262349*x^8 + 106202*x^7 + 1107150*x^6 + 1147826*x^5 - 1607860*x^4 - 6299140*x^3 - 2370356*x^2 + 8298938*x + 6534883)
 
gp: K = bnfinit(x^16 - 2*x^15 - 31*x^14 - 64*x^13 - 819*x^12 + 4002*x^11 + 27164*x^10 - 48182*x^9 - 262349*x^8 + 106202*x^7 + 1107150*x^6 + 1147826*x^5 - 1607860*x^4 - 6299140*x^3 - 2370356*x^2 + 8298938*x + 6534883, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 31 x^{14} - 64 x^{13} - 819 x^{12} + 4002 x^{11} + 27164 x^{10} - 48182 x^{9} - 262349 x^{8} + 106202 x^{7} + 1107150 x^{6} + 1147826 x^{5} - 1607860 x^{4} - 6299140 x^{3} - 2370356 x^{2} + 8298938 x + 6534883 \)

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:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{46} a^{10} + \frac{11}{46} a^{9} + \frac{5}{46} a^{8} - \frac{10}{23} a^{7} + \frac{1}{46} a^{6} - \frac{8}{23} a^{5} - \frac{11}{23} a^{4} + \frac{9}{23} a^{3} - \frac{7}{46} a^{2} - \frac{1}{23} a + \frac{4}{23}$, $\frac{1}{46} a^{11} - \frac{1}{46} a^{9} - \frac{3}{23} a^{8} + \frac{7}{23} a^{7} + \frac{19}{46} a^{6} - \frac{7}{46} a^{5} - \frac{8}{23} a^{4} - \frac{21}{46} a^{3} - \frac{17}{46} a^{2} + \frac{7}{46} a + \frac{2}{23}$, $\frac{1}{46} a^{12} + \frac{5}{46} a^{9} - \frac{2}{23} a^{8} + \frac{11}{23} a^{7} + \frac{17}{46} a^{6} + \frac{7}{23} a^{5} - \frac{10}{23} a^{4} + \frac{1}{46} a^{3} + \frac{1}{23} a - \frac{15}{46}$, $\frac{1}{46} a^{13} + \frac{5}{23} a^{9} - \frac{3}{46} a^{8} - \frac{21}{46} a^{7} - \frac{7}{23} a^{6} - \frac{9}{46} a^{5} - \frac{2}{23} a^{4} + \frac{1}{23} a^{3} - \frac{9}{46} a^{2} + \frac{9}{23} a - \frac{17}{46}$, $\frac{1}{46} a^{14} + \frac{1}{23} a^{9} - \frac{1}{23} a^{8} - \frac{21}{46} a^{7} - \frac{19}{46} a^{6} - \frac{5}{46} a^{5} - \frac{4}{23} a^{4} - \frac{5}{46} a^{3} - \frac{2}{23} a^{2} - \frac{10}{23} a + \frac{6}{23}$, $\frac{1}{2040536185951830914749606431376587778754} a^{15} - \frac{2202796795844245579892401077419780336}{1020268092975915457374803215688293889377} a^{14} - \frac{2866384130213837734800083389583588444}{1020268092975915457374803215688293889377} a^{13} - \frac{7799830723514023004539939669339324576}{1020268092975915457374803215688293889377} a^{12} + \frac{4979002551754681270614314350423900163}{1020268092975915457374803215688293889377} a^{11} + \frac{7505844887307475622945065622384437937}{1020268092975915457374803215688293889377} a^{10} + \frac{268804522881881893828043561067765968043}{2040536185951830914749606431376587778754} a^{9} + \frac{486572355871114507096317981449935756751}{2040536185951830914749606431376587778754} a^{8} - \frac{643692276585807678195311900215046942339}{2040536185951830914749606431376587778754} a^{7} - \frac{309246787698835787686924948419296312774}{1020268092975915457374803215688293889377} a^{6} + \frac{6646356322414776083086048603116914895}{18057842353556025794244304702447679458} a^{5} - \frac{198892116106770632066538232352303506133}{1020268092975915457374803215688293889377} a^{4} + \frac{401486060516901909690256782892253185849}{1020268092975915457374803215688293889377} a^{3} - \frac{383750208877826268348227201505932077341}{1020268092975915457374803215688293889377} a^{2} + \frac{260656318717297021116992698567856801817}{2040536185951830914749606431376587778754} a - \frac{305185499685735048193301805542122182109}{2040536185951830914749606431376587778754}$

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:  $11$
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:  \( 767739175044 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1194
Character table for t16n1194 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
41Data not computed