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Properties

Label 16.0.10092058240...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 11^{2}\cdot 19^{4}$
Root discriminant $15.41$
Ramified primes $2, 5, 11, 19$
Class number $1$
Class group Trivial
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 39, -66, 6, 180, -299, 156, 99, -236, 261, -238, 172, -90, 33, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 33*x^14 - 90*x^13 + 172*x^12 - 238*x^11 + 261*x^10 - 236*x^9 + 99*x^8 + 156*x^7 - 299*x^6 + 180*x^5 + 6*x^4 - 66*x^3 + 39*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^16 - 8*x^15 + 33*x^14 - 90*x^13 + 172*x^12 - 238*x^11 + 261*x^10 - 236*x^9 + 99*x^8 + 156*x^7 - 299*x^6 + 180*x^5 + 6*x^4 - 66*x^3 + 39*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 33 x^{14} - 90 x^{13} + 172 x^{12} - 238 x^{11} + 261 x^{10} - 236 x^{9} + 99 x^{8} + 156 x^{7} - 299 x^{6} + 180 x^{5} + 6 x^{4} - 66 x^{3} + 39 x^{2} - 10 x + 1 \)

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:  \(10092058240000000000=2^{16}\cdot 5^{10}\cdot 11^{2}\cdot 19^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.41$
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:  $2, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{175} a^{14} - \frac{3}{35} a^{13} - \frac{3}{175} a^{12} - \frac{54}{175} a^{11} - \frac{11}{25} a^{10} + \frac{8}{35} a^{9} - \frac{12}{175} a^{8} - \frac{17}{175} a^{7} - \frac{3}{35} a^{6} + \frac{33}{175} a^{5} + \frac{2}{35} a^{4} + \frac{1}{25} a^{3} - \frac{53}{175} a^{2} + \frac{18}{175} a + \frac{36}{175}$, $\frac{1}{14875} a^{15} - \frac{2}{875} a^{14} + \frac{492}{14875} a^{13} - \frac{557}{14875} a^{12} - \frac{288}{875} a^{11} + \frac{249}{875} a^{10} + \frac{278}{14875} a^{9} + \frac{5286}{14875} a^{8} + \frac{234}{2125} a^{7} - \frac{146}{875} a^{6} - \frac{3767}{14875} a^{5} - \frac{5153}{14875} a^{4} - \frac{1166}{14875} a^{3} - \frac{218}{595} a^{2} + \frac{639}{14875} a + \frac{2501}{14875}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{437}{175} a^{15} - \frac{3371}{175} a^{14} + \frac{13439}{175} a^{13} - \frac{7078}{35} a^{12} + \frac{12951}{35} a^{11} - \frac{84958}{175} a^{10} + \frac{89251}{175} a^{9} - \frac{77522}{175} a^{8} + \frac{3051}{25} a^{7} + \frac{73411}{175} a^{6} - \frac{107383}{175} a^{5} + \frac{45504}{175} a^{4} + \frac{15682}{175} a^{3} - \frac{23056}{175} a^{2} + \frac{9834}{175} a - \frac{1471}{175} \) (order $4$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2410.48264069 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T799:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 62 conjugacy class representatives for t16n799 are not computed
Character table for t16n799 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), 4.2.7600.1, 4.2.475.1, \(\Q(i, \sqrt{5})\), 8.0.57760000.2

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$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}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$
19.4.0.1$x^{4} - 2 x + 10$$1$$4$$0$$C_4$$[\ ]^{4}$