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Properties

Label 16.6.55165324595...4787.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,43^{3}\cdot 97^{12}$
Root discriminant $62.57$
Ramified primes $43, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1251

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6173, 17112, 22546, 6369, -11875, -12306, -12797, -9814, 1271, 4861, 1312, -363, -60, 49, -11, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 11*x^14 + 49*x^13 - 60*x^12 - 363*x^11 + 1312*x^10 + 4861*x^9 + 1271*x^8 - 9814*x^7 - 12797*x^6 - 12306*x^5 - 11875*x^4 + 6369*x^3 + 22546*x^2 + 17112*x + 6173)
 
gp: K = bnfinit(x^16 - 5*x^15 - 11*x^14 + 49*x^13 - 60*x^12 - 363*x^11 + 1312*x^10 + 4861*x^9 + 1271*x^8 - 9814*x^7 - 12797*x^6 - 12306*x^5 - 11875*x^4 + 6369*x^3 + 22546*x^2 + 17112*x + 6173, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 11 x^{14} + 49 x^{13} - 60 x^{12} - 363 x^{11} + 1312 x^{10} + 4861 x^{9} + 1271 x^{8} - 9814 x^{7} - 12797 x^{6} - 12306 x^{5} - 11875 x^{4} + 6369 x^{3} + 22546 x^{2} + 17112 x + 6173 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-55165324595664289089457824787=-\,43^{3}\cdot 97^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.57$
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:  $43, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6}$, $\frac{1}{108} a^{14} + \frac{1}{18} a^{13} - \frac{11}{54} a^{12} - \frac{7}{108} a^{11} - \frac{1}{12} a^{10} - \frac{31}{108} a^{9} - \frac{5}{54} a^{8} + \frac{5}{54} a^{7} - \frac{1}{12} a^{6} - \frac{5}{12} a^{5} + \frac{37}{108} a^{4} + \frac{11}{27} a^{3} + \frac{4}{27} a^{2} + \frac{25}{108} a + \frac{43}{108}$, $\frac{1}{263858127390306036140177866428} a^{15} + \frac{2303908441716588057806853}{2443130809169500334631276541} a^{14} - \frac{1886143928929948307964651616}{65964531847576509035044466607} a^{13} + \frac{16321009611999097918977179831}{263858127390306036140177866428} a^{12} - \frac{1611418876109325424185298963}{87952709130102012046725955476} a^{11} + \frac{63471343502517169664093291771}{263858127390306036140177866428} a^{10} - \frac{3487040532856416310000283863}{65964531847576509035044466607} a^{9} - \frac{30564484887851039977461468689}{65964531847576509035044466607} a^{8} + \frac{42866217201833725448218800379}{87952709130102012046725955476} a^{7} + \frac{11286602961513606188895944059}{29317569710034004015575318492} a^{6} - \frac{91217021555188344017556570185}{263858127390306036140177866428} a^{5} - \frac{52834838328097098373610670077}{131929063695153018070088933214} a^{4} + \frac{47322983101092914539147912949}{131929063695153018070088933214} a^{3} - \frac{10578400231958819686374688349}{263858127390306036140177866428} a^{2} - \frac{86544457570132104865184537057}{263858127390306036140177866428} a + \frac{15493880475071660193438862721}{43976354565051006023362977738}$

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:  $10$
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:  \( 387819918.535 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1251:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 34 conjugacy class representatives for t16n1251
Character table for t16n1251 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.6.35817796211947.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
43Data not computed
97Data not computed