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Properties

Label 16.16.8638056230...0625.1
Degree $16$
Signature $[16, 0]$
Discriminant $5^{12}\cdot 29^{12}$
Root discriminant $41.79$
Ramified primes $5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3299, -29130, -50036, 21110, 92315, 15240, -64854, -20855, 23394, 8490, -4769, -1610, 560, 145, -36, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 36*x^14 + 145*x^13 + 560*x^12 - 1610*x^11 - 4769*x^10 + 8490*x^9 + 23394*x^8 - 20855*x^7 - 64854*x^6 + 15240*x^5 + 92315*x^4 + 21110*x^3 - 50036*x^2 - 29130*x - 3299)
 
gp: K = bnfinit(x^16 - 5*x^15 - 36*x^14 + 145*x^13 + 560*x^12 - 1610*x^11 - 4769*x^10 + 8490*x^9 + 23394*x^8 - 20855*x^7 - 64854*x^6 + 15240*x^5 + 92315*x^4 + 21110*x^3 - 50036*x^2 - 29130*x - 3299, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 36 x^{14} + 145 x^{13} + 560 x^{12} - 1610 x^{11} - 4769 x^{10} + 8490 x^{9} + 23394 x^{8} - 20855 x^{7} - 64854 x^{6} + 15240 x^{5} + 92315 x^{4} + 21110 x^{3} - 50036 x^{2} - 29130 x - 3299 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(86380562306022715087890625=5^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.79$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{218} a^{13} - \frac{11}{109} a^{12} - \frac{49}{109} a^{11} - \frac{79}{218} a^{10} - \frac{4}{109} a^{9} - \frac{14}{109} a^{8} + \frac{47}{218} a^{7} + \frac{15}{109} a^{6} - \frac{30}{109} a^{5} - \frac{101}{218} a^{4} - \frac{23}{109} a^{3} - \frac{35}{109} a^{2} - \frac{95}{218} a - \frac{49}{109}$, $\frac{1}{218} a^{14} - \frac{37}{218} a^{12} - \frac{55}{218} a^{11} - \frac{1}{109} a^{10} - \frac{95}{218} a^{9} + \frac{85}{218} a^{8} - \frac{13}{109} a^{7} + \frac{55}{218} a^{6} + \frac{105}{218} a^{5} - \frac{44}{109} a^{4} - \frac{101}{218} a^{3} - \frac{1}{2} a^{2} - \frac{4}{109} a - \frac{85}{218}$, $\frac{1}{266522136997222} a^{15} - \frac{138591187411}{133261068498611} a^{14} + \frac{297450992080}{133261068498611} a^{13} + \frac{19028930083587}{133261068498611} a^{12} - \frac{24772543848748}{133261068498611} a^{11} + \frac{27431248056120}{133261068498611} a^{10} + \frac{56559694816649}{133261068498611} a^{9} - \frac{57864404794155}{133261068498611} a^{8} - \frac{4464448587079}{133261068498611} a^{7} + \frac{63240190886888}{133261068498611} a^{6} + \frac{41465970458812}{133261068498611} a^{5} - \frac{16866797908345}{133261068498611} a^{4} - \frac{21015308584234}{133261068498611} a^{3} - \frac{58176870688526}{133261068498611} a^{2} + \frac{47149879163234}{133261068498611} a - \frac{47524729127117}{266522136997222}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 57322151.8284 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

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 $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), 4.4.3048625.2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.3048625.1, 4.4.4205.1 x2, 4.4.725.1 x2, 8.8.9294114390625.2, 8.8.442050625.1, 8.8.9294114390625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$