↑

Properties

Label 16.8.93432784589...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{8}\cdot 17^{2}\cdot 19^{6}$
Root discriminant $64.66$
Ramified primes $2, 5, 17, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1228

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -57996, 225604, 737468, 513090, -32524, -44188, 50412, -9170, -14460, 4100, 380, -662, 148, 4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 4*x^14 + 148*x^13 - 662*x^12 + 380*x^11 + 4100*x^10 - 14460*x^9 - 9170*x^8 + 50412*x^7 - 44188*x^6 - 32524*x^5 + 513090*x^4 + 737468*x^3 + 225604*x^2 - 57996*x + 81)
 
gp: K = bnfinit(x^16 - 4*x^15 + 4*x^14 + 148*x^13 - 662*x^12 + 380*x^11 + 4100*x^10 - 14460*x^9 - 9170*x^8 + 50412*x^7 - 44188*x^6 - 32524*x^5 + 513090*x^4 + 737468*x^3 + 225604*x^2 - 57996*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 4 x^{14} + 148 x^{13} - 662 x^{12} + 380 x^{11} + 4100 x^{10} - 14460 x^{9} - 9170 x^{8} + 50412 x^{7} - 44188 x^{6} - 32524 x^{5} + 513090 x^{4} + 737468 x^{3} + 225604 x^{2} - 57996 x + 81 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(93432784589729195622400000000=2^{44}\cdot 5^{8}\cdot 17^{2}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.66$
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:  $2, 5, 17, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{4530} a^{14} + \frac{51}{755} a^{13} - \frac{659}{4530} a^{12} - \frac{101}{906} a^{11} - \frac{5}{151} a^{10} + \frac{59}{453} a^{9} - \frac{187}{906} a^{8} - \frac{164}{453} a^{7} - \frac{125}{906} a^{6} - \frac{824}{2265} a^{5} - \frac{1403}{4530} a^{4} + \frac{85}{302} a^{3} - \frac{201}{755} a^{2} + \frac{349}{2265} a - \frac{461}{1510}$, $\frac{1}{5741327382044318617711892226824023110} a^{15} + \frac{13502754975233630169166964178799}{521938852913119874337444747893093010} a^{14} - \frac{272811065660517023877505825961324081}{5741327382044318617711892226824023110} a^{13} - \frac{1184643858566762954974915707801117317}{5741327382044318617711892226824023110} a^{12} - \frac{58440568975796923169071971439971800}{574132738204431861771189222682402311} a^{11} - \frac{100790520476779520624277718697020955}{1148265476408863723542378445364804622} a^{10} + \frac{210184603424079581787090848322000955}{1148265476408863723542378445364804622} a^{9} - \frac{64399505162833090158871083725685163}{382755158802954574514126148454934874} a^{8} - \frac{366658266109772496640958336861885779}{1148265476408863723542378445364804622} a^{7} + \frac{208617038067866777660462967491075833}{637925264671590957523543580758224790} a^{6} + \frac{3825908207912117550543708079020463}{521938852913119874337444747893093010} a^{5} + \frac{211266255438823868392618983436755851}{5741327382044318617711892226824023110} a^{4} + \frac{2557305482740654545710693623245668}{9290173757353266371702090981915895} a^{3} - \frac{10210780231838364609101196017912213}{1148265476408863723542378445364804622} a^{2} + \frac{2367412018931188664728302361250690251}{5741327382044318617711892226824023110} a - \frac{70223071825456581488474723450428051}{637925264671590957523543580758224790}$

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:  $11$
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:  \( 486447791.271 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), 4.4.7600.1, 4.4.30400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.14786560000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.8.6.1$x^{8} + 57 x^{4} + 1444$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$