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Properties

Label 16.0.92600652331...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 41^{14}$
Root discriminant $86.18$
Ramified primes $5, 41$
Class number $5000$ (GRH)
Class group $[2, 50, 50]$ (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![15698896, -21746376, 15629360, -6523510, 2609089, -524611, 116867, 77070, -36239, 15109, -2638, 323, 249, -82, 31, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 31*x^14 - 82*x^13 + 249*x^12 + 323*x^11 - 2638*x^10 + 15109*x^9 - 36239*x^8 + 77070*x^7 + 116867*x^6 - 524611*x^5 + 2609089*x^4 - 6523510*x^3 + 15629360*x^2 - 21746376*x + 15698896)
 
gp: K = bnfinit(x^16 - 3*x^15 + 31*x^14 - 82*x^13 + 249*x^12 + 323*x^11 - 2638*x^10 + 15109*x^9 - 36239*x^8 + 77070*x^7 + 116867*x^6 - 524611*x^5 + 2609089*x^4 - 6523510*x^3 + 15629360*x^2 - 21746376*x + 15698896, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 31 x^{14} - 82 x^{13} + 249 x^{12} + 323 x^{11} - 2638 x^{10} + 15109 x^{9} - 36239 x^{8} + 77070 x^{7} + 116867 x^{6} - 524611 x^{5} + 2609089 x^{4} - 6523510 x^{3} + 15629360 x^{2} - 21746376 x + 15698896 \)

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:  \(9260065233133681348183837890625=5^{12}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.18$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{2480} a^{12} - \frac{33}{620} a^{11} + \frac{21}{1240} a^{10} - \frac{67}{2480} a^{9} + \frac{13}{310} a^{8} + \frac{21}{155} a^{7} - \frac{9}{496} a^{6} - \frac{2}{31} a^{5} - \frac{77}{1240} a^{4} + \frac{67}{496} a^{3} - \frac{3}{124} a^{2} + \frac{8}{155} a - \frac{3}{10}$, $\frac{1}{4960} a^{13} - \frac{1}{4960} a^{12} + \frac{11}{496} a^{11} - \frac{29}{992} a^{10} + \frac{7}{4960} a^{9} + \frac{2}{31} a^{8} - \frac{669}{4960} a^{7} - \frac{219}{992} a^{6} + \frac{603}{2480} a^{5} + \frac{621}{4960} a^{4} + \frac{85}{992} a^{3} - \frac{383}{1240} a^{2} + \frac{143}{620} a - \frac{3}{20}$, $\frac{1}{853120} a^{14} - \frac{21}{853120} a^{13} - \frac{33}{213280} a^{12} + \frac{26039}{853120} a^{11} + \frac{31583}{853120} a^{10} + \frac{24677}{426560} a^{9} - \frac{2507}{27520} a^{8} - \frac{147667}{853120} a^{7} - \frac{25243}{106640} a^{6} - \frac{23739}{853120} a^{5} - \frac{177487}{853120} a^{4} - \frac{61611}{426560} a^{3} + \frac{12889}{53320} a^{2} - \frac{5645}{21328} a - \frac{337}{1720}$, $\frac{1}{39494936236022345290924173597806080} a^{15} + \frac{950347017580644375612522199}{19747468118011172645462086798903040} a^{14} - \frac{176248347401826000457085953127}{7898987247204469058184834719561216} a^{13} - \frac{320032944431132388560659824145}{7898987247204469058184834719561216} a^{12} + \frac{490127416410320484079869752352341}{9873734059005586322731043399451520} a^{11} + \frac{1370612459778587545800718758094151}{39494936236022345290924173597806080} a^{10} + \frac{377046313066889151768371414353561}{39494936236022345290924173597806080} a^{9} + \frac{164788961560637131133441632841659}{3949493623602234529092417359780608} a^{8} - \frac{7764687297390163800474184184736881}{39494936236022345290924173597806080} a^{7} + \frac{1157102702378542739464345307526393}{7898987247204469058184834719561216} a^{6} - \frac{397260227564607119609112789851857}{2468433514751396580682760849862880} a^{5} + \frac{9085574405182898395560705325295741}{39494936236022345290924173597806080} a^{4} + \frac{3229441713682792727984015831632623}{19747468118011172645462086798903040} a^{3} - \frac{38371394924837930093127149996229}{114810861151227747938733062784320} a^{2} - \frac{1614236560267803093143085942430561}{4936867029502793161365521699725760} a + \frac{39015245260033280736581198999213}{79626887572625696151056801608480}$

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

Class group and class number

$C_{2}\times C_{50}\times C_{50}$, which has order $5000$ (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:  \( 1050930.24893 \) (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

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{41})\), 4.4.1723025.1, 4.4.68921.1, 8.8.2968815150625.1, 8.0.3043035529390625.1 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$41$41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$