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Properties

Label 16.0.37923607082...9089.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 83^{8}$
Root discriminant $108.69$
Ramified primes $17, 83$
Class number $1272$ (GRH)
Class group $[1272]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![779940859, -321982852, -12439955, -42164356, 60004187, -31253884, 15339864, -3884708, -616589, 474882, -24804, -22002, 2806, 480, -86, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 86*x^14 + 480*x^13 + 2806*x^12 - 22002*x^11 - 24804*x^10 + 474882*x^9 - 616589*x^8 - 3884708*x^7 + 15339864*x^6 - 31253884*x^5 + 60004187*x^4 - 42164356*x^3 - 12439955*x^2 - 321982852*x + 779940859)
 
gp: K = bnfinit(x^16 - 4*x^15 - 86*x^14 + 480*x^13 + 2806*x^12 - 22002*x^11 - 24804*x^10 + 474882*x^9 - 616589*x^8 - 3884708*x^7 + 15339864*x^6 - 31253884*x^5 + 60004187*x^4 - 42164356*x^3 - 12439955*x^2 - 321982852*x + 779940859, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 86 x^{14} + 480 x^{13} + 2806 x^{12} - 22002 x^{11} - 24804 x^{10} + 474882 x^{9} - 616589 x^{8} - 3884708 x^{7} + 15339864 x^{6} - 31253884 x^{5} + 60004187 x^{4} - 42164356 x^{3} - 12439955 x^{2} - 321982852 x + 779940859 \)

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:  \(379236070824193420342929192569089=17^{14}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $108.69$
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:  $17, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{34} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{15}{34} a^{4} - \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{15}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{9} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} + \frac{13}{34} a^{5} - \frac{5}{17} a^{4} - \frac{5}{34} a^{3} - \frac{3}{34} a^{2} + \frac{5}{34} a + \frac{1}{17}$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{7} + \frac{1}{34} a^{6} - \frac{2}{17} a^{5} - \frac{9}{34} a^{4} - \frac{9}{34} a^{3} - \frac{7}{34} a^{2} - \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{34} a^{11} - \frac{7}{34} a^{7} - \frac{7}{17} a^{6} + \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{5}{34} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{25874} a^{12} - \frac{9}{761} a^{11} - \frac{96}{12937} a^{10} - \frac{89}{25874} a^{9} - \frac{66}{12937} a^{8} - \frac{1372}{12937} a^{7} - \frac{6133}{25874} a^{6} - \frac{1716}{12937} a^{5} + \frac{4833}{12937} a^{4} - \frac{12069}{25874} a^{3} - \frac{201}{761} a^{2} - \frac{3103}{12937} a + \frac{6711}{25874}$, $\frac{1}{25874} a^{13} - \frac{225}{25874} a^{11} - \frac{122}{12937} a^{10} + \frac{15}{12937} a^{9} + \frac{241}{25874} a^{8} - \frac{3968}{12937} a^{7} - \frac{11875}{25874} a^{6} - \frac{239}{25874} a^{5} + \frac{1470}{12937} a^{4} - \frac{744}{12937} a^{3} + \frac{42}{761} a^{2} + \frac{2424}{12937} a + \frac{2671}{25874}$, $\frac{1}{25874} a^{14} + \frac{157}{25874} a^{11} + \frac{207}{25874} a^{10} + \frac{1}{12937} a^{9} - \frac{347}{25874} a^{8} - \frac{1486}{12937} a^{7} + \frac{599}{1522} a^{6} + \frac{12287}{25874} a^{5} - \frac{815}{25874} a^{4} + \frac{10283}{25874} a^{3} - \frac{3118}{12937} a^{2} + \frac{617}{12937} a - \frac{4110}{12937}$, $\frac{1}{6016135409189875196853709493151666747487056566} a^{15} - \frac{113973404585642271970093453942423171380871}{6016135409189875196853709493151666747487056566} a^{14} - \frac{27205142203680501428944381765026008538410}{3008067704594937598426854746575833373743528283} a^{13} + \frac{20324971453251841898353832668185028670119}{6016135409189875196853709493151666747487056566} a^{12} - \frac{1883575844255101041581448144962372700594863}{6016135409189875196853709493151666747487056566} a^{11} + \frac{47487112664876947589215333132377383540917133}{6016135409189875196853709493151666747487056566} a^{10} - \frac{41233019110031934404295274049115936122111263}{6016135409189875196853709493151666747487056566} a^{9} - \frac{24449804790070007894537430237344474309693595}{6016135409189875196853709493151666747487056566} a^{8} - \frac{447209455430905517621327906534561953169493984}{3008067704594937598426854746575833373743528283} a^{7} + \frac{1362749232351965706819888179406854191577285553}{3008067704594937598426854746575833373743528283} a^{6} - \frac{59451450271055721994173400906687006587030221}{6016135409189875196853709493151666747487056566} a^{5} + \frac{2404030850403806990048920612328596432777811583}{6016135409189875196853709493151666747487056566} a^{4} - \frac{1239437215736777954681301324145101906641117817}{3008067704594937598426854746575833373743528283} a^{3} - \frac{1493287303473269574203660296679511594575682915}{3008067704594937598426854746575833373743528283} a^{2} - \frac{227611852367973410318311153264004049916149612}{3008067704594937598426854746575833373743528283} a + \frac{1416411570979315698620396021782031195423212802}{3008067704594937598426854746575833373743528283}$

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

Class group and class number

$C_{1272}$, which has order $1272$ (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:  \( 2344259.36682 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-83}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-1411}) \), \(\Q(\sqrt{17}, \sqrt{-83})\), 4.4.4913.1, 4.0.33845657.2, 8.0.1145528497761649.3, 8.4.2826823118297.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
$83$83.8.4.1$x^{8} + 303116 x^{4} - 571787 x^{2} + 22969827364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
83.8.4.1$x^{8} + 303116 x^{4} - 571787 x^{2} + 22969827364$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$