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Properties

Label 16.4.39986964567...0000.2
Degree $16$
Signature $[4, 6]$
Discriminant $2^{24}\cdot 5^{8}\cdot 41^{3}\cdot 97^{4}$
Root discriminant $39.82$
Ramified primes $2, 5, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T905

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100409, -157894, -63513, 77740, -8750, 36534, -32718, 3878, -3452, 3622, 209, -2, -135, -46, 6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 6*x^14 - 46*x^13 - 135*x^12 - 2*x^11 + 209*x^10 + 3622*x^9 - 3452*x^8 + 3878*x^7 - 32718*x^6 + 36534*x^5 - 8750*x^4 + 77740*x^3 - 63513*x^2 - 157894*x + 100409)
 
gp: K = bnfinit(x^16 + 6*x^14 - 46*x^13 - 135*x^12 - 2*x^11 + 209*x^10 + 3622*x^9 - 3452*x^8 + 3878*x^7 - 32718*x^6 + 36534*x^5 - 8750*x^4 + 77740*x^3 - 63513*x^2 - 157894*x + 100409, 1)
 

Normalized defining polynomial

\( x^{16} + 6 x^{14} - 46 x^{13} - 135 x^{12} - 2 x^{11} + 209 x^{10} + 3622 x^{9} - 3452 x^{8} + 3878 x^{7} - 32718 x^{6} + 36534 x^{5} - 8750 x^{4} + 77740 x^{3} - 63513 x^{2} - 157894 x + 100409 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(39986964567169433600000000=2^{24}\cdot 5^{8}\cdot 41^{3}\cdot 97^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.82$
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:  $2, 5, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{65} a^{14} - \frac{22}{65} a^{13} - \frac{11}{65} a^{12} - \frac{27}{65} a^{11} - \frac{2}{13} a^{10} + \frac{6}{13} a^{9} + \frac{9}{65} a^{8} + \frac{29}{65} a^{7} - \frac{19}{65} a^{6} - \frac{28}{65} a^{5} - \frac{28}{65} a^{4} + \frac{23}{65} a^{3} + \frac{27}{65} a^{2} - \frac{27}{65} a - \frac{6}{65}$, $\frac{1}{27986550503494824435033391222001230835} a^{15} + \frac{152732118292505939020309383344630293}{27986550503494824435033391222001230835} a^{14} + \frac{1847602678192739050685088549519387414}{27986550503494824435033391222001230835} a^{13} + \frac{3146264344247411545429586123166429613}{27986550503494824435033391222001230835} a^{12} + \frac{1079486750049177269972883255653360795}{5597310100698964887006678244400246167} a^{11} + \frac{1865662602827013004831554492793564132}{5597310100698964887006678244400246167} a^{10} + \frac{10389908687256876961347351397612226659}{27986550503494824435033391222001230835} a^{9} + \frac{10796563760936948227934728188082437044}{27986550503494824435033391222001230835} a^{8} - \frac{11250806108229444676363686496564896109}{27986550503494824435033391222001230835} a^{7} + \frac{4746305245581936658181875960313122942}{27986550503494824435033391222001230835} a^{6} + \frac{8385943407562838757789269792545045472}{27986550503494824435033391222001230835} a^{5} - \frac{394777781363605130170413271367590874}{2152811577191909571925645478615479295} a^{4} + \frac{10959372141070786936559864430794823872}{27986550503494824435033391222001230835} a^{3} - \frac{8679672300523549165914008664901081347}{27986550503494824435033391222001230835} a^{2} - \frac{4371126187573148096116748127725268321}{27986550503494824435033391222001230835} a + \frac{1076060326260886692529895804441232004}{5597310100698964887006678244400246167}$

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:  $9$
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:  \( 2774362.30462 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T905:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n905
Character table for t16n905 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.104960000.1

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$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}$
5.4.2.1$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}$
41Data not computed
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$