↑

Properties

Label 16.0.92257744828...8125.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{8}\cdot 29^{5}\cdot 109^{4}$
Root discriminant $74.61$
Ramified primes $5, 13, 29, 109$
Class number $104$ (GRH)
Class group $[2, 52]$ (GRH)
Galois group 16T1581

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![128909, -292433, 325013, -76748, -43181, -35827, 52321, -2335, -2526, -3139, 1414, 14, -152, 51, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 3*x^14 + 51*x^13 - 152*x^12 + 14*x^11 + 1414*x^10 - 3139*x^9 - 2526*x^8 - 2335*x^7 + 52321*x^6 - 35827*x^5 - 43181*x^4 - 76748*x^3 + 325013*x^2 - 292433*x + 128909)
 
gp: K = bnfinit(x^16 - 6*x^15 + 3*x^14 + 51*x^13 - 152*x^12 + 14*x^11 + 1414*x^10 - 3139*x^9 - 2526*x^8 - 2335*x^7 + 52321*x^6 - 35827*x^5 - 43181*x^4 - 76748*x^3 + 325013*x^2 - 292433*x + 128909, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 3 x^{14} + 51 x^{13} - 152 x^{12} + 14 x^{11} + 1414 x^{10} - 3139 x^{9} - 2526 x^{8} - 2335 x^{7} + 52321 x^{6} - 35827 x^{5} - 43181 x^{4} - 76748 x^{3} + 325013 x^{2} - 292433 x + 128909 \)

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:  \(922577448288361543921511328125=5^{8}\cdot 13^{8}\cdot 29^{5}\cdot 109^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.61$
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:  $5, 13, 29, 109$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10} a^{14} - \frac{3}{10} a^{13} + \frac{3}{10} a^{12} + \frac{3}{10} a^{11} + \frac{2}{5} a^{10} + \frac{3}{10} a^{9} - \frac{1}{10} a^{8} - \frac{1}{2} a^{7} + \frac{1}{10} a^{4} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{10}$, $\frac{1}{4322165219474196073192733386700429761810} a^{15} + \frac{88979403590240904476742303380320698957}{2161082609737098036596366693350214880905} a^{14} + \frac{60993017028146164566326283621862336011}{2161082609737098036596366693350214880905} a^{13} + \frac{53151369121046217976473196688443694957}{2161082609737098036596366693350214880905} a^{12} + \frac{419639255234644067942673821007251950411}{864433043894839214638546677340085952362} a^{11} - \frac{109342711086499794047992384380430990699}{4322165219474196073192733386700429761810} a^{10} + \frac{167118723337121902312529579271656625542}{432216521947419607319273338670042976181} a^{9} - \frac{13808355087388563170817671705902115456}{2161082609737098036596366693350214880905} a^{8} + \frac{409468272477928532786865634767880985297}{864433043894839214638546677340085952362} a^{7} - \frac{63035828750033108903766471954958824595}{432216521947419607319273338670042976181} a^{6} - \frac{932869035845524118446341788015929721529}{4322165219474196073192733386700429761810} a^{5} + \frac{152627151007509833706370267860286572133}{4322165219474196073192733386700429761810} a^{4} - \frac{925555818157432387442204541266122203426}{2161082609737098036596366693350214880905} a^{3} - \frac{611781334612805965206650158500876658897}{2161082609737098036596366693350214880905} a^{2} + \frac{1662938522720647880558670782486177391897}{4322165219474196073192733386700429761810} a - \frac{1479509826349162503829577861965391778917}{4322165219474196073192733386700429761810}$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$ (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:  \( 3377432.05839 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1581:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 94 conjugacy class representatives for t16n1581 are not computed
Character table for t16n1581 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.13355225.1, 4.0.2725.1, 4.4.122525.1, 8.0.178362034800625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
$109$109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$