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Properties

Label 16.0.76422304165...0625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 89^{14}$
Root discriminant $113.55$
Ramified primes $5, 89$
Class number $16900$ (GRH)
Class group $[65, 260]$ (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![42852464, 102320, 4757904, 7513276, -3244715, 677858, 833586, -400144, 108574, 890, -4918, 2702, -514, 48, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 18*x^14 + 48*x^13 - 514*x^12 + 2702*x^11 - 4918*x^10 + 890*x^9 + 108574*x^8 - 400144*x^7 + 833586*x^6 + 677858*x^5 - 3244715*x^4 + 7513276*x^3 + 4757904*x^2 + 102320*x + 42852464)
 
gp: K = bnfinit(x^16 - 6*x^15 + 18*x^14 + 48*x^13 - 514*x^12 + 2702*x^11 - 4918*x^10 + 890*x^9 + 108574*x^8 - 400144*x^7 + 833586*x^6 + 677858*x^5 - 3244715*x^4 + 7513276*x^3 + 4757904*x^2 + 102320*x + 42852464, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 18 x^{14} + 48 x^{13} - 514 x^{12} + 2702 x^{11} - 4918 x^{10} + 890 x^{9} + 108574 x^{8} - 400144 x^{7} + 833586 x^{6} + 677858 x^{5} - 3244715 x^{4} + 7513276 x^{3} + 4757904 x^{2} + 102320 x + 42852464 \)

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:  \(764223041656422427087574156640625=5^{8}\cdot 89^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $113.55$
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:  $5, 89$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{44} a^{8} + \frac{1}{44} a^{7} + \frac{1}{22} a^{6} - \frac{2}{11} a^{5} - \frac{3}{22} a^{4} + \frac{2}{11} a^{3} - \frac{9}{44} a^{2} + \frac{21}{44} a - \frac{5}{22}$, $\frac{1}{44} a^{9} + \frac{1}{44} a^{7} - \frac{5}{22} a^{6} + \frac{1}{22} a^{5} - \frac{2}{11} a^{4} - \frac{17}{44} a^{3} - \frac{7}{22} a^{2} - \frac{9}{44} a + \frac{5}{22}$, $\frac{1}{88} a^{10} - \frac{1}{88} a^{9} - \frac{1}{88} a^{8} - \frac{1}{44} a^{7} + \frac{1}{11} a^{6} - \frac{2}{11} a^{5} + \frac{3}{88} a^{4} - \frac{35}{88} a^{3} - \frac{43}{88} a^{2} + \frac{5}{44} a + \frac{4}{11}$, $\frac{1}{176} a^{11} - \frac{1}{88} a^{9} + \frac{1}{176} a^{8} - \frac{3}{44} a^{7} + \frac{43}{176} a^{5} - \frac{3}{44} a^{4} - \frac{1}{88} a^{3} + \frac{19}{176} a^{2} - \frac{9}{22} a + \frac{9}{44}$, $\frac{1}{880} a^{12} + \frac{1}{440} a^{11} + \frac{1}{220} a^{10} - \frac{9}{880} a^{9} - \frac{1}{110} a^{8} + \frac{13}{110} a^{7} + \frac{19}{880} a^{6} + \frac{9}{88} a^{5} + \frac{3}{22} a^{4} - \frac{131}{880} a^{3} - \frac{47}{220} a^{2} + \frac{41}{220} a + \frac{6}{55}$, $\frac{1}{880} a^{13} + \frac{3}{880} a^{10} - \frac{1}{88} a^{9} - \frac{109}{880} a^{7} + \frac{3}{220} a^{6} - \frac{1}{44} a^{5} - \frac{151}{880} a^{4} - \frac{53}{440} a^{3} + \frac{7}{44} a^{2} - \frac{19}{110} a + \frac{8}{55}$, $\frac{1}{5052080} a^{14} - \frac{527}{5052080} a^{13} + \frac{5}{505208} a^{12} + \frac{1181}{631510} a^{11} + \frac{14319}{5052080} a^{10} - \frac{1715}{252604} a^{9} + \frac{15403}{2526040} a^{8} + \frac{4315}{1010416} a^{7} - \frac{249717}{2526040} a^{6} + \frac{251711}{1263020} a^{5} - \frac{405899}{5052080} a^{4} - \frac{256921}{631510} a^{3} + \frac{1911933}{5052080} a^{2} - \frac{226517}{1263020} a - \frac{44429}{114820}$, $\frac{1}{11409495773457349669993951604167040} a^{15} - \frac{6761104141138400439045961}{570474788672867483499697580208352} a^{14} - \frac{437513905082924199385498603687}{5704747886728674834996975802083520} a^{13} + \frac{158503341267140213501009782645}{570474788672867483499697580208352} a^{12} + \frac{8730493184618208736422803316867}{5704747886728674834996975802083520} a^{11} - \frac{25371803095514184005055896192767}{5704747886728674834996975802083520} a^{10} + \frac{33019339287525545651977850633439}{5704747886728674834996975802083520} a^{9} - \frac{2634970464988593532820763163127}{518613444248061348636088709280320} a^{8} - \frac{617548928978993808110747366939663}{5704747886728674834996975802083520} a^{7} + \frac{83854130785955874208078579412961}{2852373943364337417498487901041760} a^{6} + \frac{316546906335426060489101107577517}{5704747886728674834996975802083520} a^{5} + \frac{1365611185744517723839519107759367}{5704747886728674834996975802083520} a^{4} + \frac{2832036300293788897602696802069}{18858670699929503586766862155648} a^{3} - \frac{780253123447457296790306499240449}{5704747886728674834996975802083520} a^{2} + \frac{36218106285725143876820198428849}{259306722124030674318044354640160} a - \frac{126873730730892470202034539364515}{285237394336433741749848790104176}$

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

Class group and class number

$C_{65}\times C_{260}$, which has order $16900$ (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:  \( 8274413.16122 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{445}) \), \(\Q(\sqrt{5}, \sqrt{89})\), 4.4.704969.1, 4.4.17624225.2, 8.8.310613306850625.1, 8.0.1105783372388225.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$89$89.8.7.1$x^{8} - 89$$8$$1$$7$$C_8$$[\ ]_{8}$
89.8.7.1$x^{8} - 89$$8$$1$$7$$C_8$$[\ ]_{8}$