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Properties

Label 16.6.55165324595...4787.2
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![7744, 32384, 28272, 94968, 7052, 59112, -16425, 3563, -6013, -3330, 915, -221, 62, 110, -27, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 27*x^14 + 110*x^13 + 62*x^12 - 221*x^11 + 915*x^10 - 3330*x^9 - 6013*x^8 + 3563*x^7 - 16425*x^6 + 59112*x^5 + 7052*x^4 + 94968*x^3 + 28272*x^2 + 32384*x + 7744)
 
gp: K = bnfinit(x^16 - 4*x^15 - 27*x^14 + 110*x^13 + 62*x^12 - 221*x^11 + 915*x^10 - 3330*x^9 - 6013*x^8 + 3563*x^7 - 16425*x^6 + 59112*x^5 + 7052*x^4 + 94968*x^3 + 28272*x^2 + 32384*x + 7744, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 27 x^{14} + 110 x^{13} + 62 x^{12} - 221 x^{11} + 915 x^{10} - 3330 x^{9} - 6013 x^{8} + 3563 x^{7} - 16425 x^{6} + 59112 x^{5} + 7052 x^{4} + 94968 x^{3} + 28272 x^{2} + 32384 x + 7744 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{12} + \frac{1}{6} a^{11} - \frac{1}{4} a^{10} - \frac{1}{3} a^{9} - \frac{1}{6} a^{8} - \frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{1}{3} a^{5} - \frac{5}{12} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{24} a^{13} + \frac{5}{24} a^{11} + \frac{1}{12} a^{10} + \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{24} a^{7} + \frac{5}{12} a^{6} + \frac{11}{24} a^{5} + \frac{7}{24} a^{4} + \frac{1}{8} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{48} a^{14} + \frac{1}{48} a^{12} - \frac{1}{8} a^{11} + \frac{3}{8} a^{10} - \frac{5}{48} a^{9} + \frac{7}{48} a^{8} + \frac{7}{24} a^{7} + \frac{23}{48} a^{6} - \frac{3}{16} a^{5} + \frac{23}{48} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3419893237312077958592750283936} a^{15} - \frac{157709166642321191092378483}{213743327332004872412046892746} a^{14} + \frac{19268900892949330527312105799}{1139964412437359319530916761312} a^{13} + \frac{1596131557797271777306007545}{51816564201698150887768943696} a^{12} + \frac{103125996075052124539294313673}{569982206218679659765458380656} a^{11} + \frac{525886179205200676039990684475}{3419893237312077958592750283936} a^{10} + \frac{1705329149602503883993270172879}{3419893237312077958592750283936} a^{9} - \frac{225927035570226269006033484083}{1709946618656038979296375141968} a^{8} + \frac{392937297028295095497535564521}{1139964412437359319530916761312} a^{7} - \frac{348355744630266322667481825747}{1139964412437359319530916761312} a^{6} - \frac{483324403730868923194386693005}{3419893237312077958592750283936} a^{5} + \frac{141154926545257809658886709471}{284991103109339829882729190328} a^{4} + \frac{183902686076596691945843117227}{854973309328019489648187570984} a^{3} + \frac{16076315161536768422404683175}{142495551554669914941364595164} a^{2} - \frac{74461262525089759802509939345}{213743327332004872412046892746} a - \frac{294317679353132806446991910}{9715605787818403291456676943}$

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