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Properties

Label 16.0.41154791040...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 3^{12}\cdot 5^{10}\cdot 11^{2}$
Root discriminant $16.82$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T799

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19, -142, 505, -1204, 2192, -3172, 3748, -3680, 3007, -2046, 1159, -540, 211, -70, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 22*x^14 - 70*x^13 + 211*x^12 - 540*x^11 + 1159*x^10 - 2046*x^9 + 3007*x^8 - 3680*x^7 + 3748*x^6 - 3172*x^5 + 2192*x^4 - 1204*x^3 + 505*x^2 - 142*x + 19)
 
gp: K = bnfinit(x^16 - 6*x^15 + 22*x^14 - 70*x^13 + 211*x^12 - 540*x^11 + 1159*x^10 - 2046*x^9 + 3007*x^8 - 3680*x^7 + 3748*x^6 - 3172*x^5 + 2192*x^4 - 1204*x^3 + 505*x^2 - 142*x + 19, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 22 x^{14} - 70 x^{13} + 211 x^{12} - 540 x^{11} + 1159 x^{10} - 2046 x^{9} + 3007 x^{8} - 3680 x^{7} + 3748 x^{6} - 3172 x^{5} + 2192 x^{4} - 1204 x^{3} + 505 x^{2} - 142 x + 19 \)

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:  \(41154791040000000000=2^{16}\cdot 3^{12}\cdot 5^{10}\cdot 11^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.82$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{38} a^{14} - \frac{7}{19} a^{13} - \frac{1}{2} a^{12} - \frac{9}{19} a^{11} - \frac{3}{19} a^{10} - \frac{9}{19} a^{9} + \frac{17}{38} a^{8} + \frac{1}{19} a^{7} + \frac{5}{19} a^{6} + \frac{7}{19} a^{4} - \frac{8}{19} a^{3} - \frac{6}{19} a^{2} + \frac{5}{19} a - \frac{1}{2}$, $\frac{1}{11546284306} a^{15} + \frac{4480362}{5773142153} a^{14} + \frac{855755857}{11546284306} a^{13} - \frac{380698487}{5773142153} a^{12} - \frac{1088063087}{5773142153} a^{11} + \frac{1228047447}{5773142153} a^{10} + \frac{40926671}{312061738} a^{9} - \frac{1888905154}{5773142153} a^{8} - \frac{1553776641}{5773142153} a^{7} - \frac{2404311491}{5773142153} a^{6} + \frac{189481782}{5773142153} a^{5} + \frac{2434937668}{5773142153} a^{4} + \frac{1235864564}{5773142153} a^{3} + \frac{1082816945}{5773142153} a^{2} + \frac{4666895727}{11546284306} a + \frac{16519848}{303849587}$

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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{236458391}{312061738} a^{15} + \frac{650057234}{156030869} a^{14} - \frac{4555022795}{312061738} a^{13} + \frac{7141908675}{156030869} a^{12} - \frac{21387086779}{156030869} a^{11} + \frac{53188416505}{156030869} a^{10} - \frac{221117039811}{312061738} a^{9} + \frac{186842304118}{156030869} a^{8} - \frac{262472437119}{156030869} a^{7} + \frac{304180507109}{156030869} a^{6} - \frac{291233781465}{156030869} a^{5} + \frac{229212535905}{156030869} a^{4} - \frac{144074969721}{156030869} a^{3} + \frac{69667115127}{156030869} a^{2} - \frac{48462915681}{312061738} a + \frac{230433617}{8212151} \) (order $6$)
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:  \( 5861.5743076 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.2.3600.1, 4.2.400.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.12960000.2

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.8.8.2$x^{8} + 2 x^{7} + 8 x^{2} + 48$$2$$4$$8$$C_2^2:C_4$$[2, 2]^{4}$
3Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$