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Properties

Label 16.0.19680337591...1329.5
Degree $16$
Signature $[0, 8]$
Discriminant $17^{14}\cdot 43^{8}$
Root discriminant $78.23$
Ramified primes $17, 43$
Class number $984$ (GRH)
Class group $[984]$ (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![16275257, 18597494, 5655015, -4147460, -1606091, 449552, 369868, 17336, 64057, -5402, -4132, -854, 630, 4, -18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 18*x^14 + 4*x^13 + 630*x^12 - 854*x^11 - 4132*x^10 - 5402*x^9 + 64057*x^8 + 17336*x^7 + 369868*x^6 + 449552*x^5 - 1606091*x^4 - 4147460*x^3 + 5655015*x^2 + 18597494*x + 16275257)
 
gp: K = bnfinit(x^16 - 4*x^15 - 18*x^14 + 4*x^13 + 630*x^12 - 854*x^11 - 4132*x^10 - 5402*x^9 + 64057*x^8 + 17336*x^7 + 369868*x^6 + 449552*x^5 - 1606091*x^4 - 4147460*x^3 + 5655015*x^2 + 18597494*x + 16275257, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 18 x^{14} + 4 x^{13} + 630 x^{12} - 854 x^{11} - 4132 x^{10} - 5402 x^{9} + 64057 x^{8} + 17336 x^{7} + 369868 x^{6} + 449552 x^{5} - 1606091 x^{4} - 4147460 x^{3} + 5655015 x^{2} + 18597494 x + 16275257 \)

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:  \(1968033759133442969054057291329=17^{14}\cdot 43^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.23$
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:  $17, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{34} a^{8} - \frac{1}{17} a^{7} + \frac{3}{17} a^{6} + \frac{7}{17} a^{5} - \frac{15}{34} a^{4} - \frac{7}{17} a^{3} - \frac{11}{34} a^{2} - \frac{15}{34} a + \frac{1}{34}$, $\frac{1}{34} a^{9} + \frac{1}{17} a^{7} - \frac{4}{17} a^{6} + \frac{13}{34} a^{5} - \frac{5}{17} a^{4} - \frac{5}{34} a^{3} - \frac{3}{34} a^{2} + \frac{5}{34} a + \frac{1}{17}$, $\frac{1}{34} a^{10} - \frac{2}{17} a^{7} + \frac{1}{34} a^{6} - \frac{2}{17} a^{5} - \frac{9}{34} a^{4} - \frac{9}{34} a^{3} - \frac{7}{34} a^{2} - \frac{1}{17} a - \frac{1}{17}$, $\frac{1}{34} a^{11} - \frac{7}{34} a^{7} - \frac{7}{17} a^{6} + \frac{13}{34} a^{5} - \frac{1}{34} a^{4} + \frac{5}{34} a^{3} - \frac{6}{17} a^{2} + \frac{3}{17} a + \frac{2}{17}$, $\frac{1}{3026} a^{12} + \frac{1}{3026} a^{11} + \frac{21}{3026} a^{10} + \frac{14}{1513} a^{9} + \frac{6}{1513} a^{8} + \frac{1103}{3026} a^{7} - \frac{11}{1513} a^{6} - \frac{44}{1513} a^{5} + \frac{543}{1513} a^{4} + \frac{532}{1513} a^{3} + \frac{695}{1513} a^{2} + \frac{1353}{3026} a - \frac{643}{3026}$, $\frac{1}{3026} a^{13} + \frac{10}{1513} a^{11} + \frac{7}{3026} a^{10} - \frac{8}{1513} a^{9} + \frac{23}{3026} a^{8} + \frac{1011}{3026} a^{7} - \frac{211}{1513} a^{6} + \frac{676}{1513} a^{5} + \frac{434}{1513} a^{4} + \frac{74}{1513} a^{3} - \frac{393}{3026} a^{2} - \frac{553}{1513} a - \frac{25}{178}$, $\frac{1}{3026} a^{14} - \frac{13}{3026} a^{11} + \frac{9}{3026} a^{10} - \frac{3}{3026} a^{9} - \frac{15}{1513} a^{8} - \frac{205}{1513} a^{7} - \frac{789}{3026} a^{6} - \frac{199}{1513} a^{5} - \frac{373}{1513} a^{4} + \frac{511}{1513} a^{3} - \frac{302}{1513} a^{2} + \frac{720}{1513} a + \frac{133}{3026}$, $\frac{1}{111287290191662718724266999996589537114} a^{15} + \frac{564390740660427533409114454624919}{6546311187744865807309823529211149242} a^{14} - \frac{14766302528182132057951523548956417}{111287290191662718724266999996589537114} a^{13} + \frac{10563593508324220685882313798757201}{111287290191662718724266999996589537114} a^{12} + \frac{633601506627714091481106800004274745}{55643645095831359362133499998294768557} a^{11} + \frac{109892205926930098428371558441750788}{55643645095831359362133499998294768557} a^{10} + \frac{31779934401600964460547602861570589}{111287290191662718724266999996589537114} a^{9} + \frac{549084678314700099779163537006006614}{55643645095831359362133499998294768557} a^{8} - \frac{8356074227205997822417820716854151825}{55643645095831359362133499998294768557} a^{7} - \frac{185286402778727943562609029191842600}{625209495458779318675657303351626613} a^{6} - \frac{4276500122864727894601173429118648273}{55643645095831359362133499998294768557} a^{5} + \frac{16520604471770043616142937574769132805}{55643645095831359362133499998294768557} a^{4} - \frac{25534666340509760470934104268378895527}{111287290191662718724266999996589537114} a^{3} - \frac{52251070493041129499471030612445310927}{111287290191662718724266999996589537114} a^{2} + \frac{21246999644948481189187882909747499816}{55643645095831359362133499998294768557} a - \frac{50497990936508466829473956729871308217}{111287290191662718724266999996589537114}$

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

Class group and class number

$C_{984}$, which has order $984$ (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:  \( 238836.43919 \) (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

\(\Q(\sqrt{-43}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-731}) \), \(\Q(\sqrt{17}, \sqrt{-43})\), 4.4.4913.1, 4.0.9084137.2, 8.0.82521545034769.3, 8.4.758716206377.1 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 ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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}$
$43$43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$