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Properties

Label 16.8.28020861582...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}$
Root discriminant $79.98$
Ramified primes $2, 5, 101, 641$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1161

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1556119, 3510750, 1945660, -1927690, 677094, -860760, 38600, 36470, 18321, 18540, -940, 250, -766, -110, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 110*x^13 - 766*x^12 + 250*x^11 - 940*x^10 + 18540*x^9 + 18321*x^8 + 36470*x^7 + 38600*x^6 - 860760*x^5 + 677094*x^4 - 1927690*x^3 + 1945660*x^2 + 3510750*x - 1556119)
 
gp: K = bnfinit(x^16 - 110*x^13 - 766*x^12 + 250*x^11 - 940*x^10 + 18540*x^9 + 18321*x^8 + 36470*x^7 + 38600*x^6 - 860760*x^5 + 677094*x^4 - 1927690*x^3 + 1945660*x^2 + 3510750*x - 1556119, 1)
 

Normalized defining polynomial

\( x^{16} - 110 x^{13} - 766 x^{12} + 250 x^{11} - 940 x^{10} + 18540 x^{9} + 18321 x^{8} + 36470 x^{7} + 38600 x^{6} - 860760 x^{5} + 677094 x^{4} - 1927690 x^{3} + 1945660 x^{2} + 3510750 x - 1556119 \)

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:  \(2802086158223343616000000000000=2^{28}\cdot 5^{12}\cdot 101^{4}\cdot 641^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.98$
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:  $2, 5, 101, 641$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{77302664776109350024885001028936440669203585547488579} a^{15} + \frac{22347800967690272353564506307240323411895126931380669}{77302664776109350024885001028936440669203585547488579} a^{14} - \frac{32483855852798760852126946012766911953233283315774586}{77302664776109350024885001028936440669203585547488579} a^{13} - \frac{9047393733710844378309512101786795017870736202607850}{77302664776109350024885001028936440669203585547488579} a^{12} - \frac{35674629079673767717639497697078204369059448463292905}{77302664776109350024885001028936440669203585547488579} a^{11} + \frac{13468403085995028068288811434828801888930691466962275}{77302664776109350024885001028936440669203585547488579} a^{10} - \frac{5610058268123463816229386131563130457683715675992117}{77302664776109350024885001028936440669203585547488579} a^{9} - \frac{35605130757938632186578717193264115108017873258018263}{77302664776109350024885001028936440669203585547488579} a^{8} + \frac{16760655012876215660736836989634250967908699850361015}{77302664776109350024885001028936440669203585547488579} a^{7} + \frac{11336261205667012360870517054529860486675656509994674}{77302664776109350024885001028936440669203585547488579} a^{6} + \frac{33679900044920339254703848987793554651596312529452557}{77302664776109350024885001028936440669203585547488579} a^{5} + \frac{36774326018571532030502703159324632206811138820991549}{77302664776109350024885001028936440669203585547488579} a^{4} - \frac{7856715058296365929271835493029868483592882085137003}{77302664776109350024885001028936440669203585547488579} a^{3} + \frac{12698454385030883801982561475528548449646522908397147}{77302664776109350024885001028936440669203585547488579} a^{2} + \frac{31133431723933666267646919939167792566911788633842168}{77302664776109350024885001028936440669203585547488579} a - \frac{21212412592552368242771122242593767372246210030422223}{77302664776109350024885001028936440669203585547488579}$

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:  \( 2156184367.32 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1161:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 43 conjugacy class representatives for t16n1161
Character table for t16n1161 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 8.8.6464000000.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
101Data not computed
641Data not computed