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Properties

Label 16.16.6028340169...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{14}\cdot 61^{14}$
Root discriminant $149.20$
Ramified primes $5, 61$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $OD_{16}.C_2$ (as 16T40)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-486704, 18646992, 438205392, -583498754, -150618369, 343311742, -14339755, -49762742, 5316819, 2956884, -394695, -84296, 12506, 1148, -182, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 182*x^14 + 1148*x^13 + 12506*x^12 - 84296*x^11 - 394695*x^10 + 2956884*x^9 + 5316819*x^8 - 49762742*x^7 - 14339755*x^6 + 343311742*x^5 - 150618369*x^4 - 583498754*x^3 + 438205392*x^2 + 18646992*x - 486704)
 
gp: K = bnfinit(x^16 - 6*x^15 - 182*x^14 + 1148*x^13 + 12506*x^12 - 84296*x^11 - 394695*x^10 + 2956884*x^9 + 5316819*x^8 - 49762742*x^7 - 14339755*x^6 + 343311742*x^5 - 150618369*x^4 - 583498754*x^3 + 438205392*x^2 + 18646992*x - 486704, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 182 x^{14} + 1148 x^{13} + 12506 x^{12} - 84296 x^{11} - 394695 x^{10} + 2956884 x^{9} + 5316819 x^{8} - 49762742 x^{7} - 14339755 x^{6} + 343311742 x^{5} - 150618369 x^{4} - 583498754 x^{3} + 438205392 x^{2} + 18646992 x - 486704 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(60283401692879138764114019775390625=5^{14}\cdot 61^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.20$
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:  $5, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{424} a^{14} - \frac{3}{106} a^{13} + \frac{11}{212} a^{12} + \frac{15}{424} a^{11} - \frac{25}{212} a^{10} + \frac{7}{106} a^{9} - \frac{41}{424} a^{8} + \frac{87}{424} a^{7} - \frac{43}{212} a^{6} + \frac{5}{53} a^{5} + \frac{19}{53} a^{4} + \frac{77}{424} a^{3} - \frac{31}{212} a^{2} + \frac{31}{106} a + \frac{12}{53}$, $\frac{1}{17666435497663689832534728542756212954655736835144} a^{15} - \frac{4553677219001487669248309788233349063564570083}{8833217748831844916267364271378106477327868417572} a^{14} - \frac{281833498976679430911136384475301353840886010947}{8833217748831844916267364271378106477327868417572} a^{13} - \frac{35365012162302691346548625924107716033529407079}{17666435497663689832534728542756212954655736835144} a^{12} + \frac{221034001521461099237233586288467480251919596095}{17666435497663689832534728542756212954655736835144} a^{11} + \frac{1506634846483929785710417323852715141007655175369}{17666435497663689832534728542756212954655736835144} a^{10} + \frac{51305609829282763937671749907220765145046622882}{2208304437207961229066841067844526619331967104393} a^{9} - \frac{1020032569973556822512941624316119154867546093235}{8833217748831844916267364271378106477327868417572} a^{8} + \frac{215202908477860996655895479340209856013203967179}{8833217748831844916267364271378106477327868417572} a^{7} + \frac{5206850440452076550997581707488684980536846726045}{17666435497663689832534728542756212954655736835144} a^{6} + \frac{562347074964933603329847714590516825291588384273}{17666435497663689832534728542756212954655736835144} a^{5} + \frac{7767361657502211355482222120761581015271848259793}{17666435497663689832534728542756212954655736835144} a^{4} + \frac{2274080438721085595752862330462694821021926932599}{17666435497663689832534728542756212954655736835144} a^{3} + \frac{4373373490329245194745048267465994115980690403801}{8833217748831844916267364271378106477327868417572} a^{2} + \frac{257472295756307079410077659224742893109612480933}{4416608874415922458133682135689053238663934208786} a + \frac{1067925386673945787003515488197214773477731989507}{2208304437207961229066841067844526619331967104393}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $15$
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:  \( 796441295513 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}.C_2$ (as 16T40):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $OD_{16}.C_2$
Character table for $OD_{16}.C_2$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{305}) \), \(\Q(\sqrt{5}) \), 4.4.28372625.1, 4.4.28372625.2, \(\Q(\sqrt{5}, \sqrt{61})\), 8.8.805005849390625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$61$61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.2$x^{8} - 244$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$