↑

Properties

Label 16.0.63215880456...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 29^{2}\cdot 89^{2}\cdot 149^{8}$
Root discriminant $72.87$
Ramified primes $5, 29, 89, 149$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T860

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![551, -7017, 36474, -99582, 160574, -42651, -39606, 45319, -1224, -7304, 2540, 105, -327, 85, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 7*x^14 + 85*x^13 - 327*x^12 + 105*x^11 + 2540*x^10 - 7304*x^9 - 1224*x^8 + 45319*x^7 - 39606*x^6 - 42651*x^5 + 160574*x^4 - 99582*x^3 + 36474*x^2 - 7017*x + 551)
 
gp: K = bnfinit(x^16 - 6*x^15 + 7*x^14 + 85*x^13 - 327*x^12 + 105*x^11 + 2540*x^10 - 7304*x^9 - 1224*x^8 + 45319*x^7 - 39606*x^6 - 42651*x^5 + 160574*x^4 - 99582*x^3 + 36474*x^2 - 7017*x + 551, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 7 x^{14} + 85 x^{13} - 327 x^{12} + 105 x^{11} + 2540 x^{10} - 7304 x^{9} - 1224 x^{8} + 45319 x^{7} - 39606 x^{6} - 42651 x^{5} + 160574 x^{4} - 99582 x^{3} + 36474 x^{2} - 7017 x + 551 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(632158804568484064343109765625=5^{8}\cdot 29^{2}\cdot 89^{2}\cdot 149^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.87$
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:  $5, 29, 89, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{183} a^{14} + \frac{74}{183} a^{13} - \frac{15}{61} a^{12} - \frac{7}{61} a^{11} - \frac{27}{61} a^{10} + \frac{29}{61} a^{9} + \frac{47}{183} a^{8} + \frac{89}{183} a^{7} + \frac{26}{61} a^{6} + \frac{20}{61} a^{5} + \frac{22}{61} a^{4} - \frac{15}{61} a^{3} - \frac{10}{183} a^{2} - \frac{2}{183} a - \frac{41}{183}$, $\frac{1}{12501000229317278146314433973781537} a^{15} + \frac{1443851694776883790914784336709}{12501000229317278146314433973781537} a^{14} - \frac{2054972680097773872653124744465891}{4167000076439092715438144657927179} a^{13} + \frac{1650188522430219705687099902190689}{4167000076439092715438144657927179} a^{12} - \frac{1836592124369056816039010002867166}{4167000076439092715438144657927179} a^{11} - \frac{1585119513974870751316014448064961}{4167000076439092715438144657927179} a^{10} - \frac{4128671410905338737266279366979867}{12501000229317278146314433973781537} a^{9} + \frac{5407754763250067771184438193810979}{12501000229317278146314433973781537} a^{8} + \frac{1451068839053748617461779452842640}{4167000076439092715438144657927179} a^{7} + \frac{335382520594875728080333674551818}{4167000076439092715438144657927179} a^{6} + \frac{1108943321229369324288134030024272}{4167000076439092715438144657927179} a^{5} + \frac{1583036488147334477877997272082450}{4167000076439092715438144657927179} a^{4} + \frac{4019629904283860329268448748577459}{12501000229317278146314433973781537} a^{3} + \frac{931708479099074412931140263193811}{12501000229317278146314433973781537} a^{2} + \frac{3710870177799484363360113604215409}{12501000229317278146314433973781537} a + \frac{745505015631896610495081410321364}{4167000076439092715438144657927179}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $7$
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:  \( 136482759.028 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T860:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 53 conjugacy class representatives for t16n860 are not computed
Character table for t16n860 is not computed

Intermediate fields

\(\Q(\sqrt{149}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{745}) \), 4.0.111005.1 x2, 4.0.3725.1 x2, \(\Q(\sqrt{5}, \sqrt{149})\), 8.0.308052750625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
$89$89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
149Data not computed