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Properties

Label 16.0.82432069367...0721.1
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 29^{12}$
Root discriminant $85.56$
Ramified primes $13, 29$
Class number $144$ (GRH)
Class group $[12, 12]$ (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![58841, 67135, 226780, 235277, 55705, -30189, 62333, -1665, -13310, 7567, 537, -1492, 246, 98, -23, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 23*x^14 + 98*x^13 + 246*x^12 - 1492*x^11 + 537*x^10 + 7567*x^9 - 13310*x^8 - 1665*x^7 + 62333*x^6 - 30189*x^5 + 55705*x^4 + 235277*x^3 + 226780*x^2 + 67135*x + 58841)
 
gp: K = bnfinit(x^16 - 4*x^15 - 23*x^14 + 98*x^13 + 246*x^12 - 1492*x^11 + 537*x^10 + 7567*x^9 - 13310*x^8 - 1665*x^7 + 62333*x^6 - 30189*x^5 + 55705*x^4 + 235277*x^3 + 226780*x^2 + 67135*x + 58841, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 23 x^{14} + 98 x^{13} + 246 x^{12} - 1492 x^{11} + 537 x^{10} + 7567 x^{9} - 13310 x^{8} - 1665 x^{7} + 62333 x^{6} - 30189 x^{5} + 55705 x^{4} + 235277 x^{3} + 226780 x^{2} + 67135 x + 58841 \)

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:  \(8243206936713178643875538610721=13^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.56$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{214} a^{11} + \frac{31}{214} a^{10} - \frac{21}{107} a^{9} + \frac{15}{107} a^{8} - \frac{59}{214} a^{7} + \frac{95}{214} a^{6} - \frac{43}{214} a^{5} - \frac{7}{107} a^{4} + \frac{46}{107} a^{3} + \frac{81}{214} a^{2} - \frac{23}{107} a - \frac{53}{107}$, $\frac{1}{187892} a^{12} - \frac{3}{187892} a^{11} + \frac{25547}{187892} a^{10} + \frac{495}{187892} a^{9} - \frac{5486}{46973} a^{8} - \frac{467}{187892} a^{7} + \frac{35225}{93946} a^{6} - \frac{39961}{187892} a^{5} + \frac{13945}{46973} a^{4} + \frac{40395}{187892} a^{3} - \frac{29015}{187892} a^{2} + \frac{5233}{46973} a + \frac{54001}{187892}$, $\frac{1}{187892} a^{13} + \frac{19}{46973} a^{11} + \frac{19691}{93946} a^{10} + \frac{15539}{187892} a^{9} + \frac{15355}{187892} a^{8} + \frac{68171}{187892} a^{7} + \frac{7203}{187892} a^{6} - \frac{2643}{187892} a^{5} - \frac{93419}{187892} a^{4} - \frac{22394}{46973} a^{3} - \frac{61723}{187892} a^{2} - \frac{27195}{187892} a - \frac{51351}{187892}$, $\frac{1}{187892} a^{14} + \frac{25}{46973} a^{11} + \frac{43321}{187892} a^{10} + \frac{40073}{187892} a^{9} - \frac{13061}{187892} a^{8} - \frac{68811}{187892} a^{7} - \frac{91477}{187892} a^{6} + \frac{39193}{187892} a^{5} - \frac{4469}{46973} a^{4} + \frac{91395}{187892} a^{3} + \frac{11041}{187892} a^{2} + \frac{81331}{187892} a + \frac{20996}{46973}$, $\frac{1}{898475858278298229509516} a^{15} - \frac{437842322051491383}{224618964569574557377379} a^{14} + \frac{301097766959295809}{898475858278298229509516} a^{13} + \frac{1161943397582461709}{898475858278298229509516} a^{12} + \frac{59230506947847117809}{449237929139149114754758} a^{11} - \frac{55287801168186627136513}{449237929139149114754758} a^{10} + \frac{9852131711567395658023}{898475858278298229509516} a^{9} - \frac{18610268156461832248238}{224618964569574557377379} a^{8} + \frac{322021190944133432739911}{898475858278298229509516} a^{7} + \frac{31231255002870947761389}{449237929139149114754758} a^{6} - \frac{143774650833095696779673}{449237929139149114754758} a^{5} + \frac{72809421594111953072666}{224618964569574557377379} a^{4} - \frac{91223597500882117170915}{449237929139149114754758} a^{3} - \frac{382975842475772387294621}{898475858278298229509516} a^{2} + \frac{62197362177657049978031}{898475858278298229509516} a - \frac{115690758258603403812037}{449237929139149114754758}$

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

Class group and class number

$C_{12}\times C_{12}$, which has order $144$ (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:  \( 2985208.11128 \) (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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{377}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{13}, \sqrt{29})\), 4.4.4121741.1 x2, 4.4.317057.1 x2, 4.0.1847677.1, 4.0.2197.1, 8.8.16988748871081.1, 8.0.3413910296329.2, 8.0.2871098559212689.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.1$x^{4} - 29$$4$$1$$3$$C_4$$[\ ]_{4}$