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Properties

Label 16.0.18919427965...0000.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{10}\cdot 29^{6}\cdot 89^{6}$
Root discriminant $104.07$
Ramified primes $2, 5, 29, 89$
Class number $32$ (GRH)
Class group $[2, 2, 8]$ (GRH)
Galois group 16T790

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![266392369, 147746204, 27780227, 5646760, 3321886, -1339930, 219389, 186018, 31863, -5158, 2835, 1218, -286, 22, 31, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 31*x^14 + 22*x^13 - 286*x^12 + 1218*x^11 + 2835*x^10 - 5158*x^9 + 31863*x^8 + 186018*x^7 + 219389*x^6 - 1339930*x^5 + 3321886*x^4 + 5646760*x^3 + 27780227*x^2 + 147746204*x + 266392369)
 
gp: K = bnfinit(x^16 - 8*x^15 + 31*x^14 + 22*x^13 - 286*x^12 + 1218*x^11 + 2835*x^10 - 5158*x^9 + 31863*x^8 + 186018*x^7 + 219389*x^6 - 1339930*x^5 + 3321886*x^4 + 5646760*x^3 + 27780227*x^2 + 147746204*x + 266392369, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 31 x^{14} + 22 x^{13} - 286 x^{12} + 1218 x^{11} + 2835 x^{10} - 5158 x^{9} + 31863 x^{8} + 186018 x^{7} + 219389 x^{6} - 1339930 x^{5} + 3321886 x^{4} + 5646760 x^{3} + 27780227 x^{2} + 147746204 x + 266392369 \)

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:  \(189194279657145152947840000000000=2^{16}\cdot 5^{10}\cdot 29^{6}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.07$
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:  $2, 5, 29, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{13136343} a^{14} - \frac{336338}{13136343} a^{13} + \frac{165491}{1194213} a^{12} + \frac{683842}{13136343} a^{11} - \frac{1826498}{13136343} a^{10} + \frac{307580}{4378781} a^{9} - \frac{118397}{13136343} a^{8} - \frac{2694854}{13136343} a^{7} + \frac{3762079}{13136343} a^{6} + \frac{206411}{13136343} a^{5} + \frac{774689}{4378781} a^{4} - \frac{579103}{1194213} a^{3} - \frac{1724708}{4378781} a^{2} - \frac{1215627}{4378781} a + \frac{4178918}{13136343}$, $\frac{1}{109711231582197822353461366878348968782895389479183} a^{15} + \frac{1048088625226436755023763156480395757375075}{109711231582197822353461366878348968782895389479183} a^{14} - \frac{488899322735529880161052108249295031219778339839}{3324582775218115828892768693283302084330163317551} a^{13} - \frac{8889786050989319388543886117829123066315142064279}{109711231582197822353461366878348968782895389479183} a^{12} - \frac{4186984727153410979349440464890285667810087226906}{109711231582197822353461366878348968782895389479183} a^{11} + \frac{4299572782911431777103680503746834608930428912165}{36570410527399274117820455626116322927631796493061} a^{10} - \frac{12905723570735434675711578852788847505404916022587}{109711231582197822353461366878348968782895389479183} a^{9} + \frac{3060689565491947297316934860426181636951550973}{97175581560848381181099527793046030808587590327} a^{8} - \frac{56066092676869352991470482639121019131073154996}{1924758448810488111464234506637701206717462973319} a^{7} + \frac{10288769733285874564861273898240882943348923693531}{36570410527399274117820455626116322927631796493061} a^{6} + \frac{16473491994939063408348120194712478156639756191368}{36570410527399274117820455626116322927631796493061} a^{5} - \frac{1627431975694731467468387565798088601886548174706}{3324582775218115828892768693283302084330163317551} a^{4} + \frac{16702134939751574484956561214522672433337154601654}{109711231582197822353461366878348968782895389479183} a^{3} + \frac{21559344784387867778430885109117737242156528199719}{109711231582197822353461366878348968782895389479183} a^{2} + \frac{4127744212051834431769483052349603000914203865611}{109711231582197822353461366878348968782895389479183} a - \frac{155309550537050601328567944056724098847858068921}{524934122402860394035700319992100329104762629087}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, which has order $32$ (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:  \( 123798183.997 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T790:

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 t16n790 are not computed
Character table for t16n790 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.64525.1, 4.0.35600.3, 4.0.11600.1, 8.0.1065849760000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.0.1$x^{4} - x + 27$$1$$4$$0$$C_4$$[\ ]^{4}$
89.4.2.2$x^{4} - 89 x^{2} + 47526$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$