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Properties

Label 16.16.5860999329...0000.1
Degree $16$
Signature $[16, 0]$
Discriminant $2^{4}\cdot 5^{8}\cdot 6229^{5}$
Root discriminant $40.78$
Ramified primes $2, 5, 6229$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1862

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1721, -6151, -9387, 62713, -84521, 15244, 53331, -35629, -7383, 12698, -1232, -1899, 410, 127, -36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 36*x^14 + 127*x^13 + 410*x^12 - 1899*x^11 - 1232*x^10 + 12698*x^9 - 7383*x^8 - 35629*x^7 + 53331*x^6 + 15244*x^5 - 84521*x^4 + 62713*x^3 - 9387*x^2 - 6151*x + 1721)
 
gp: K = bnfinit(x^16 - 3*x^15 - 36*x^14 + 127*x^13 + 410*x^12 - 1899*x^11 - 1232*x^10 + 12698*x^9 - 7383*x^8 - 35629*x^7 + 53331*x^6 + 15244*x^5 - 84521*x^4 + 62713*x^3 - 9387*x^2 - 6151*x + 1721, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 36 x^{14} + 127 x^{13} + 410 x^{12} - 1899 x^{11} - 1232 x^{10} + 12698 x^{9} - 7383 x^{8} - 35629 x^{7} + 53331 x^{6} + 15244 x^{5} - 84521 x^{4} + 62713 x^{3} - 9387 x^{2} - 6151 x + 1721 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(58609993297236388431250000=2^{4}\cdot 5^{8}\cdot 6229^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.78$
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:  $2, 5, 6229$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{380} a^{14} - \frac{127}{380} a^{13} + \frac{43}{380} a^{12} + \frac{9}{190} a^{11} + \frac{51}{380} a^{10} + \frac{11}{76} a^{9} - \frac{51}{380} a^{8} + \frac{67}{380} a^{7} - \frac{33}{95} a^{6} - \frac{36}{95} a^{5} + \frac{1}{4} a^{4} - \frac{3}{19} a^{3} - \frac{9}{95} a^{2} - \frac{63}{380} a + \frac{109}{380}$, $\frac{1}{2593165132600} a^{15} - \frac{12963649}{13648237540} a^{14} - \frac{4899035251}{10987987850} a^{13} - \frac{606840863231}{2593165132600} a^{12} + \frac{1690728549}{19497482200} a^{11} - \frac{290720333289}{1296582566300} a^{10} + \frac{23048340818}{324145641575} a^{9} - \frac{7558843043}{129658256630} a^{8} + \frac{16864730901}{370452161800} a^{7} + \frac{202248978283}{648291283150} a^{6} - \frac{759628956553}{2593165132600} a^{5} + \frac{79000224419}{518633026520} a^{4} + \frac{19768299863}{92613040450} a^{3} + \frac{1008930257}{10584347480} a^{2} - \frac{31845465423}{185226080900} a - \frac{70549896387}{2593165132600}$

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:  $15$
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:  \( 89201372.7514 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1862:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 73728
The 152 conjugacy class representatives for t16n1862 are not computed
Character table for t16n1862 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.24250275625.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
6229Data not computed