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Properties

Label 16.0.14917095564...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{20}\cdot 5^{8}\cdot 41^{4}\cdot 359^{2}$
Root discriminant $28.08$
Ramified primes $2, 5, 41, 359$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1701

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1205, -765, -1391, 240, -594, 465, 1057, -388, 587, -185, 191, -114, 24, -18, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - 18*x^13 + 24*x^12 - 114*x^11 + 191*x^10 - 185*x^9 + 587*x^8 - 388*x^7 + 1057*x^6 + 465*x^5 - 594*x^4 + 240*x^3 - 1391*x^2 - 765*x + 1205)
 
gp: K = bnfinit(x^16 + 4*x^14 - 18*x^13 + 24*x^12 - 114*x^11 + 191*x^10 - 185*x^9 + 587*x^8 - 388*x^7 + 1057*x^6 + 465*x^5 - 594*x^4 + 240*x^3 - 1391*x^2 - 765*x + 1205, 1)
 

Normalized defining polynomial

\( x^{16} + 4 x^{14} - 18 x^{13} + 24 x^{12} - 114 x^{11} + 191 x^{10} - 185 x^{9} + 587 x^{8} - 388 x^{7} + 1057 x^{6} + 465 x^{5} - 594 x^{4} + 240 x^{3} - 1391 x^{2} - 765 x + 1205 \)

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:  \(149170955649433600000000=2^{20}\cdot 5^{8}\cdot 41^{4}\cdot 359^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.08$
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:  $2, 5, 41, 359$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{384926244022721015190863925} a^{15} + \frac{5800416782231452670389807}{384926244022721015190863925} a^{14} + \frac{5438140288116281450716976}{128308748007573671730287975} a^{13} + \frac{7705788105351752118240776}{128308748007573671730287975} a^{12} + \frac{1683350099782104937357594}{76985248804544203038172785} a^{11} - \frac{153597338125874486740379699}{384926244022721015190863925} a^{10} - \frac{185810278346367066322926727}{384926244022721015190863925} a^{9} - \frac{8127578339459595368412824}{384926244022721015190863925} a^{8} + \frac{12731996034579522241418929}{34993294911156455926442175} a^{7} + \frac{22837091248680577084718629}{76985248804544203038172785} a^{6} + \frac{81252873230563953696191747}{384926244022721015190863925} a^{5} - \frac{119395164897026348801704331}{384926244022721015190863925} a^{4} - \frac{8886395835907166445965236}{384926244022721015190863925} a^{3} - \frac{33928250049936191281917579}{128308748007573671730287975} a^{2} - \frac{3894958777926865882336369}{15397049760908840607634557} a + \frac{1696744086725185135889564}{25661749601514734346057595}$

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:  \( -1 \) (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:  \( 184497.022146 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1701:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8192
The 119 conjugacy class representatives for t16n1701 are not computed
Character table for t16n1701 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.1025.1, 8.0.67240000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.20.12$x^{8} + 4 x^{7} + 12 x^{6} + 2 x^{4} + 24 x^{2} + 20$$4$$2$$20$$C_2^3: C_4$$[2, 2, 3, 7/2]^{2}$
$5$5.2.1.1$x^{2} - 5$$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
359Data not computed