↑

Properties

Label 16.0.12809387541...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{10}\cdot 13^{6}\cdot 101^{6}$
Root discriminant $57.11$
Ramified primes $2, 5, 13, 101$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group 16T1189

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![682429, -1086192, 1167463, -974647, 411054, -43761, 30905, -23268, -24317, 18611, 729, -2718, 280, 165, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 165*x^13 + 280*x^12 - 2718*x^11 + 729*x^10 + 18611*x^9 - 24317*x^8 - 23268*x^7 + 30905*x^6 - 43761*x^5 + 411054*x^4 - 974647*x^3 + 1167463*x^2 - 1086192*x + 682429)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 165*x^13 + 280*x^12 - 2718*x^11 + 729*x^10 + 18611*x^9 - 24317*x^8 - 23268*x^7 + 30905*x^6 - 43761*x^5 + 411054*x^4 - 974647*x^3 + 1167463*x^2 - 1086192*x + 682429, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} + 165 x^{13} + 280 x^{12} - 2718 x^{11} + 729 x^{10} + 18611 x^{9} - 24317 x^{8} - 23268 x^{7} + 30905 x^{6} - 43761 x^{5} + 411054 x^{4} - 974647 x^{3} + 1167463 x^{2} - 1086192 x + 682429 \)

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:  \(12809387541505655522500000000=2^{8}\cdot 5^{10}\cdot 13^{6}\cdot 101^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.11$
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:  $2, 5, 13, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{245257115032697962062895866172066143590763} a^{15} + \frac{11299814452016330811390301926448665594960}{81752371677565987354298622057355381196921} a^{14} + \frac{34384655776968070018042317567334743219929}{245257115032697962062895866172066143590763} a^{13} + \frac{115331913811327399338436871978500481243477}{245257115032697962062895866172066143590763} a^{12} - \frac{87744951806930167520835123347766768769843}{245257115032697962062895866172066143590763} a^{11} + \frac{88565722666300817963756509146012034552288}{245257115032697962062895866172066143590763} a^{10} - \frac{1756348965211575253784669925191497168098}{81752371677565987354298622057355381196921} a^{9} + \frac{16525488538346567774587754523098921725261}{81752371677565987354298622057355381196921} a^{8} + \frac{115925094847017248796802231213999905250669}{245257115032697962062895866172066143590763} a^{7} - \frac{13542181773931252203531711207176439660896}{245257115032697962062895866172066143590763} a^{6} - \frac{103476031618958703873062866124938570735094}{245257115032697962062895866172066143590763} a^{5} + \frac{100207418988788656864062321451339522066184}{245257115032697962062895866172066143590763} a^{4} + \frac{22183951852696927328010380614424368999247}{81752371677565987354298622057355381196921} a^{3} + \frac{26328715252561554586029934978589588950927}{245257115032697962062895866172066143590763} a^{2} + \frac{9499449468051153363742653648810712248467}{245257115032697962062895866172066143590763} a + \frac{2311379770222520812949421550194948009612}{7432033788869635214027147459759580108811}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 1830320.84754 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1189:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1189 are not computed
Character table for t16n1189 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.0.1077480625.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$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}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$13$13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
13.8.0.1$x^{8} + 4 x^{2} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$