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Properties

Label 16.0.57606497003...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 11^{8}$
Root discriminant $54.33$
Ramified primes $2, 3, 5, 11$
Class number $1536$ (GRH)
Class group $[2, 2, 2, 8, 24]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1094031, -500742, 975099, -293202, 337471, -65514, 60702, -5336, 5507, -278, 998, -368, 126, 24, -4, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 4*x^14 + 24*x^13 + 126*x^12 - 368*x^11 + 998*x^10 - 278*x^9 + 5507*x^8 - 5336*x^7 + 60702*x^6 - 65514*x^5 + 337471*x^4 - 293202*x^3 + 975099*x^2 - 500742*x + 1094031)
 
gp: K = bnfinit(x^16 - 4*x^15 - 4*x^14 + 24*x^13 + 126*x^12 - 368*x^11 + 998*x^10 - 278*x^9 + 5507*x^8 - 5336*x^7 + 60702*x^6 - 65514*x^5 + 337471*x^4 - 293202*x^3 + 975099*x^2 - 500742*x + 1094031, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 4 x^{14} + 24 x^{13} + 126 x^{12} - 368 x^{11} + 998 x^{10} - 278 x^{9} + 5507 x^{8} - 5336 x^{7} + 60702 x^{6} - 65514 x^{5} + 337471 x^{4} - 293202 x^{3} + 975099 x^{2} - 500742 x + 1094031 \)

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:  \(5760649700315136000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.33$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(197,·)$, $\chi_{1320}(769,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(661,·)$, $\chi_{1320}(857,·)$, $\chi_{1320}(901,·)$, $\chi_{1320}(353,·)$, $\chi_{1320}(1189,·)$, $\chi_{1320}(1253,·)$, $\chi_{1320}(593,·)$, $\chi_{1320}(617,·)$, $\chi_{1320}(109,·)$, $\chi_{1320}(241,·)$, $\chi_{1320}(1013,·)$, $\chi_{1320}(1277,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{19} a^{11} - \frac{5}{19} a^{10} + \frac{7}{19} a^{9} + \frac{9}{19} a^{8} + \frac{5}{19} a^{7} + \frac{6}{19} a^{6} - \frac{2}{19} a^{5} + \frac{5}{19} a^{4} + \frac{2}{19} a^{3} - \frac{9}{19} a^{2} - \frac{2}{19} a + \frac{1}{19}$, $\frac{1}{57} a^{12} + \frac{1}{57} a^{10} + \frac{25}{57} a^{9} + \frac{4}{19} a^{8} + \frac{4}{19} a^{7} - \frac{10}{57} a^{6} - \frac{8}{19} a^{5} + \frac{9}{19} a^{4} + \frac{1}{57} a^{3} + \frac{10}{57} a^{2} - \frac{3}{19} a + \frac{8}{19}$, $\frac{1}{57} a^{13} + \frac{1}{57} a^{11} + \frac{25}{57} a^{10} + \frac{4}{19} a^{9} + \frac{4}{19} a^{8} - \frac{10}{57} a^{7} - \frac{8}{19} a^{6} + \frac{9}{19} a^{5} + \frac{1}{57} a^{4} + \frac{10}{57} a^{3} - \frac{3}{19} a^{2} + \frac{8}{19} a$, $\frac{1}{52023762805047} a^{14} - \frac{322540314367}{52023762805047} a^{13} - \frac{45147098300}{17341254268349} a^{12} - \frac{20742399496}{912697593071} a^{11} - \frac{12338905110029}{52023762805047} a^{10} - \frac{19258732930636}{52023762805047} a^{9} + \frac{11587999349018}{52023762805047} a^{8} - \frac{21693218868023}{52023762805047} a^{7} - \frac{7267668743120}{52023762805047} a^{6} - \frac{23446325917676}{52023762805047} a^{5} - \frac{7280313240731}{17341254268349} a^{4} - \frac{12003183469238}{52023762805047} a^{3} - \frac{24710866679140}{52023762805047} a^{2} - \frac{8102194368328}{17341254268349} a + \frac{7968726511604}{17341254268349}$, $\frac{1}{21858540439792451341539813} a^{15} + \frac{1574126830}{662380013327043980046661} a^{14} + \frac{36784008103751602516840}{7286180146597483780513271} a^{13} + \frac{11328851044601095439268}{7286180146597483780513271} a^{12} + \frac{236903843690395027167608}{21858540439792451341539813} a^{11} - \frac{272154057203458938509143}{1987140039981131940139983} a^{10} + \frac{6234748941316013494396402}{21858540439792451341539813} a^{9} + \frac{1943707027006310302242820}{7286180146597483780513271} a^{8} - \frac{2058110187134127877239608}{21858540439792451341539813} a^{7} + \frac{3627895489698210230961677}{21858540439792451341539813} a^{6} - \frac{5923138187375710864580645}{21858540439792451341539813} a^{5} - \frac{47219756729736321978205}{21858540439792451341539813} a^{4} + \frac{10306679681614560805779883}{21858540439792451341539813} a^{3} - \frac{2146939206221654124589330}{21858540439792451341539813} a^{2} + \frac{1057626076337168715312506}{7286180146597483780513271} a + \frac{3379088564389689710142701}{7286180146597483780513271}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8}\times C_{24}$, which has order $1536$ (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:  \( 16694.393243512957 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{-22}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{-22})\), \(\Q(\sqrt{10}, \sqrt{-22})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{10}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-55})\), 4.0.136125.2, 4.4.72000.1, \(\Q(\zeta_{15})^+\), 4.0.8712000.5, 8.0.37480960000.9, 8.0.75898944000000.86, 8.0.75898944000000.178, 8.0.18530015625.1, 8.0.75898944000000.119, 8.0.75898944000000.146, 8.8.5184000000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$