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Properties

Label 16.4.27728016466...0000.1
Degree $16$
Signature $[4, 6]$
Discriminant $2^{28}\cdot 5^{8}\cdot 71^{4}\cdot 101^{4}$
Root discriminant $69.21$
Ramified primes $2, 5, 71, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1275

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-43021, -77336, 252166, 333106, -29840, 251926, 219686, -174208, 43507, -978, -2134, -140, 146, -70, 24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 24*x^14 - 70*x^13 + 146*x^12 - 140*x^11 - 2134*x^10 - 978*x^9 + 43507*x^8 - 174208*x^7 + 219686*x^6 + 251926*x^5 - 29840*x^4 + 333106*x^3 + 252166*x^2 - 77336*x - 43021)
 
gp: K = bnfinit(x^16 - 6*x^15 + 24*x^14 - 70*x^13 + 146*x^12 - 140*x^11 - 2134*x^10 - 978*x^9 + 43507*x^8 - 174208*x^7 + 219686*x^6 + 251926*x^5 - 29840*x^4 + 333106*x^3 + 252166*x^2 - 77336*x - 43021, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 24 x^{14} - 70 x^{13} + 146 x^{12} - 140 x^{11} - 2134 x^{10} - 978 x^{9} + 43507 x^{8} - 174208 x^{7} + 219686 x^{6} + 251926 x^{5} - 29840 x^{4} + 333106 x^{3} + 252166 x^{2} - 77336 x - 43021 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(277280164669720467865600000000=2^{28}\cdot 5^{8}\cdot 71^{4}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.21$
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:  $2, 5, 71, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{995} a^{14} - \frac{66}{995} a^{13} - \frac{39}{995} a^{12} + \frac{133}{995} a^{11} + \frac{82}{199} a^{10} + \frac{187}{995} a^{9} + \frac{458}{995} a^{8} - \frac{82}{995} a^{7} + \frac{77}{995} a^{6} - \frac{381}{995} a^{5} + \frac{37}{995} a^{4} + \frac{26}{995} a^{3} + \frac{242}{995} a^{2} - \frac{136}{995} a - \frac{223}{995}$, $\frac{1}{47226707427006488902191693732046875454204505} a^{15} - \frac{2665587877678995568745247492088092256774}{6746672489572355557455956247435267922029215} a^{14} + \frac{4074770725370247238224606488318504134140203}{47226707427006488902191693732046875454204505} a^{13} + \frac{3738828933028963715305200984475494893676702}{47226707427006488902191693732046875454204505} a^{12} - \frac{19371648206796954541074087228587610055308863}{47226707427006488902191693732046875454204505} a^{11} - \frac{6420617256874793415319368918516146199936767}{47226707427006488902191693732046875454204505} a^{10} - \frac{1821833355837689208104456470717699741272931}{9445341485401297780438338746409375090840901} a^{9} - \frac{3777961951246288201207931511502349437866116}{47226707427006488902191693732046875454204505} a^{8} - \frac{18287938056663350603382829449220387498761006}{47226707427006488902191693732046875454204505} a^{7} - \frac{14254433339310790871834112333810055162961044}{47226707427006488902191693732046875454204505} a^{6} - \frac{438122028813225815199811423242826579713386}{1349334497914471111491191249487053584405843} a^{5} - \frac{395656330817777132895378121747292607573799}{47226707427006488902191693732046875454204505} a^{4} - \frac{126068726384545217104744180704813436337812}{47226707427006488902191693732046875454204505} a^{3} + \frac{19660882422334569330552561450092420992553226}{47226707427006488902191693732046875454204505} a^{2} + \frac{21513754342570310288989838525388582450970998}{47226707427006488902191693732046875454204505} a + \frac{15202522095842428279931573494007620355670302}{47226707427006488902191693732046875454204505}$

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:  $9$
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:  \( 368785319.596 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1275:

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 t16n1275
Character table for t16n1275 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.28400.1, 8.4.1303400960000.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
$71$71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
71.4.2.1$x^{4} + 1491 x^{2} + 609961$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101Data not computed