↑

Properties

Label 16.0.28254727797...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{8}\cdot 29^{6}\cdot 41^{6}$
Root discriminant $45.00$
Ramified primes $2, 5, 29, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T486)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1200629, -1179401, 320785, -9358, 12986, -28040, -2180, -4303, 9642, 549, -397, -75, -170, 75, 4, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 4*x^14 + 75*x^13 - 170*x^12 - 75*x^11 - 397*x^10 + 549*x^9 + 9642*x^8 - 4303*x^7 - 2180*x^6 - 28040*x^5 + 12986*x^4 - 9358*x^3 + 320785*x^2 - 1179401*x + 1200629)
 
gp: K = bnfinit(x^16 - 6*x^15 + 4*x^14 + 75*x^13 - 170*x^12 - 75*x^11 - 397*x^10 + 549*x^9 + 9642*x^8 - 4303*x^7 - 2180*x^6 - 28040*x^5 + 12986*x^4 - 9358*x^3 + 320785*x^2 - 1179401*x + 1200629, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 4 x^{14} + 75 x^{13} - 170 x^{12} - 75 x^{11} - 397 x^{10} + 549 x^{9} + 9642 x^{8} - 4303 x^{7} - 2180 x^{6} - 28040 x^{5} + 12986 x^{4} - 9358 x^{3} + 320785 x^{2} - 1179401 x + 1200629 \)

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:  \(282547277972780436100000000=2^{8}\cdot 5^{8}\cdot 29^{6}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.00$
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:  $2, 5, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{24911} a^{14} + \frac{4279}{24911} a^{13} + \frac{6383}{24911} a^{12} - \frac{8639}{24911} a^{11} + \frac{9538}{24911} a^{10} - \frac{4187}{24911} a^{9} + \frac{536}{24911} a^{8} + \frac{7988}{24911} a^{7} - \frac{6208}{24911} a^{6} - \frac{6964}{24911} a^{5} + \frac{6674}{24911} a^{4} + \frac{11682}{24911} a^{3} - \frac{310}{24911} a^{2} - \frac{112}{859} a + \frac{412}{859}$, $\frac{1}{919429683088120069945399819501942686662179} a^{15} + \frac{2222849272658410855452192929540246763}{919429683088120069945399819501942686662179} a^{14} - \frac{14415513390257371610053432833569259637705}{919429683088120069945399819501942686662179} a^{13} + \frac{14303479477072776519136683855912879214189}{31704471830624829998117235155239402988351} a^{12} - \frac{186768264195877826472756183069804281528145}{919429683088120069945399819501942686662179} a^{11} - \frac{272843321256307676048277814925397941394297}{919429683088120069945399819501942686662179} a^{10} - \frac{252429732164294668973929244899192381966176}{919429683088120069945399819501942686662179} a^{9} + \frac{355535425533214960932481457667928283413769}{919429683088120069945399819501942686662179} a^{8} - \frac{2006470866240369593919952140805037160534}{83584516644374551813218165409267516969289} a^{7} + \frac{167740366371745572331514706769676907929837}{919429683088120069945399819501942686662179} a^{6} - \frac{21233992070365791603720733476759585186015}{48391035952006319470810516815891720350641} a^{5} + \frac{434663933883051131807141674590732889742149}{919429683088120069945399819501942686662179} a^{4} - \frac{67877351449474969398309619392394358742145}{919429683088120069945399819501942686662179} a^{3} - \frac{16809276321812835360476985554582215305594}{919429683088120069945399819501942686662179} a^{2} + \frac{4311164912636784351292657485666141374775}{31704471830624829998117235155239402988351} a + \frac{304260610455882624502504886356002411}{1942556940789463268066738260844274431}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 1265957.42846 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T486):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1025.1, 4.0.29725.2, 4.4.725.1, 8.0.883575625.2

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$