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Properties

Label 16.0.48736115900...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{8}$
Root discriminant $26.18$
Ramified primes $5, 41$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T157)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9801, -41202, 66681, -39468, -21139, 47194, -23126, -6054, 11282, -3714, -841, 919, -214, -13, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 21*x^14 - 13*x^13 - 214*x^12 + 919*x^11 - 841*x^10 - 3714*x^9 + 11282*x^8 - 6054*x^7 - 23126*x^6 + 47194*x^5 - 21139*x^4 - 39468*x^3 + 66681*x^2 - 41202*x + 9801)
 
gp: K = bnfinit(x^16 - 7*x^15 + 21*x^14 - 13*x^13 - 214*x^12 + 919*x^11 - 841*x^10 - 3714*x^9 + 11282*x^8 - 6054*x^7 - 23126*x^6 + 47194*x^5 - 21139*x^4 - 39468*x^3 + 66681*x^2 - 41202*x + 9801, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 21 x^{14} - 13 x^{13} - 214 x^{12} + 919 x^{11} - 841 x^{10} - 3714 x^{9} + 11282 x^{8} - 6054 x^{7} - 23126 x^{6} + 47194 x^{5} - 21139 x^{4} - 39468 x^{3} + 66681 x^{2} - 41202 x + 9801 \)

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:  \(48736115900396728515625=5^{14}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.18$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} - \frac{1}{3} a^{12} - \frac{4}{9} a^{11} - \frac{4}{9} a^{10} + \frac{4}{9} a^{9} + \frac{2}{9} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} - \frac{1}{3} a^{5} + \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{77590941645085562490687} a^{15} - \frac{735380798818416969334}{77590941645085562490687} a^{14} + \frac{351422356430163616915}{8621215738342840276743} a^{13} - \frac{10596778918813057980865}{77590941645085562490687} a^{12} + \frac{3758147097184625022710}{77590941645085562490687} a^{11} - \frac{882314563010459796236}{77590941645085562490687} a^{10} - \frac{7670720454139706043511}{77590941645085562490687} a^{9} - \frac{3655654785464273574467}{8621215738342840276743} a^{8} + \frac{15253750427358959417372}{77590941645085562490687} a^{7} + \frac{782436368380036512528}{2873738579447613425581} a^{6} + \frac{8378743405733295676951}{77590941645085562490687} a^{5} - \frac{15812662798901732742908}{77590941645085562490687} a^{4} - \frac{30863258538757282949215}{77590941645085562490687} a^{3} - \frac{10838921707979519211245}{25863647215028520830229} a^{2} + \frac{3092095070962560738143}{8621215738342840276743} a + \frac{108450587224534869335}{261248961767964856871}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( \frac{284560717286636053}{38544928785437437899} a^{15} - \frac{1543431475348409546}{38544928785437437899} a^{14} + \frac{3609022863917952616}{38544928785437437899} a^{13} + \frac{1672862797833276146}{38544928785437437899} a^{12} - \frac{19218123458178755144}{12848309595145812633} a^{11} + \frac{171354011597994012389}{38544928785437437899} a^{10} + \frac{18337999188524074510}{38544928785437437899} a^{9} - \frac{997911003713745953789}{38544928785437437899} a^{8} + \frac{1664122451679774490550}{38544928785437437899} a^{7} + \frac{705690516808865493712}{38544928785437437899} a^{6} - \frac{5284745126839083074780}{38544928785437437899} a^{5} + \frac{1796752793196559891736}{12848309595145812633} a^{4} + \frac{1766809674187435298038}{38544928785437437899} a^{3} - \frac{8142838640875326118592}{38544928785437437899} a^{2} + \frac{2215402801703996951401}{12848309595145812633} a - \frac{210589846276674768008}{4282769865048604211} \) (order $10$)
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:  \( 93871.1737576 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T157):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 8.0.5384453125.1 x2, 8.0.26265625.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
Degree 16 siblings: data not computed
Degree 32 siblings: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
5Data not computed
41Data not computed