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Properties

Label 16.0.89791815397...000.22
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}$
Root discriminant $55.86$
Ramified primes $2, 3, 5, 13$
Class number $1536$ (GRH)
Class group $[4, 4, 4, 24]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, -14580, 9072, -20088, 43695, -34272, 32116, -39664, 18768, -4076, 3614, -588, 371, -56, 30, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 30*x^14 - 56*x^13 + 371*x^12 - 588*x^11 + 3614*x^10 - 4076*x^9 + 18768*x^8 - 39664*x^7 + 32116*x^6 - 34272*x^5 + 43695*x^4 - 20088*x^3 + 9072*x^2 - 14580*x + 6561)
 
gp: K = bnfinit(x^16 - 4*x^15 + 30*x^14 - 56*x^13 + 371*x^12 - 588*x^11 + 3614*x^10 - 4076*x^9 + 18768*x^8 - 39664*x^7 + 32116*x^6 - 34272*x^5 + 43695*x^4 - 20088*x^3 + 9072*x^2 - 14580*x + 6561, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 30 x^{14} - 56 x^{13} + 371 x^{12} - 588 x^{11} + 3614 x^{10} - 4076 x^{9} + 18768 x^{8} - 39664 x^{7} + 32116 x^{6} - 34272 x^{5} + 43695 x^{4} - 20088 x^{3} + 9072 x^{2} - 14580 x + 6561 \)

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:  \(8979181539709000089600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.86$
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:  $2, 3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1560=2^{3}\cdot 3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1560}(1,·)$, $\chi_{1560}(1091,·)$, $\chi_{1560}(389,·)$, $\chi_{1560}(961,·)$, $\chi_{1560}(521,·)$, $\chi_{1560}(1559,·)$, $\chi_{1560}(1039,·)$, $\chi_{1560}(131,·)$, $\chi_{1560}(1429,·)$, $\chi_{1560}(599,·)$, $\chi_{1560}(79,·)$, $\chi_{1560}(989,·)$, $\chi_{1560}(1171,·)$, $\chi_{1560}(1481,·)$, $\chi_{1560}(571,·)$, $\chi_{1560}(469,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{3}$, $\frac{1}{45} a^{10} - \frac{1}{45} a^{9} + \frac{2}{45} a^{8} - \frac{2}{15} a^{7} + \frac{1}{15} a^{6} + \frac{2}{15} a^{5} - \frac{16}{45} a^{4} + \frac{7}{45} a^{3} - \frac{8}{45} a^{2} + \frac{7}{15} a - \frac{1}{5}$, $\frac{1}{135} a^{11} - \frac{4}{135} a^{9} + \frac{2}{45} a^{8} + \frac{4}{45} a^{7} + \frac{1}{15} a^{6} + \frac{4}{27} a^{5} + \frac{4}{15} a^{4} - \frac{56}{135} a^{3} + \frac{1}{45} a^{2} - \frac{2}{15} a - \frac{2}{5}$, $\frac{1}{270} a^{12} + \frac{1}{135} a^{10} - \frac{1}{45} a^{8} + \frac{1}{15} a^{7} + \frac{19}{135} a^{6} - \frac{1}{15} a^{5} + \frac{59}{135} a^{4} + \frac{1}{3} a^{3} - \frac{2}{15} a^{2} + \frac{4}{15} a + \frac{3}{10}$, $\frac{1}{270} a^{13} + \frac{1}{135} a^{9} + \frac{1}{45} a^{8} + \frac{7}{135} a^{7} - \frac{2}{15} a^{6} - \frac{2}{45} a^{5} + \frac{1}{15} a^{4} - \frac{7}{135} a^{3} + \frac{11}{45} a^{2} + \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{28350} a^{14} + \frac{1}{2025} a^{13} - \frac{1}{675} a^{12} - \frac{7}{2025} a^{11} - \frac{1}{405} a^{10} - \frac{28}{675} a^{9} - \frac{272}{14175} a^{8} + \frac{563}{14175} a^{7} + \frac{41}{675} a^{6} + \frac{262}{2025} a^{5} - \frac{442}{2025} a^{4} - \frac{49}{675} a^{3} + \frac{67}{3150} a^{2} - \frac{1}{175} a - \frac{83}{175}$, $\frac{1}{4642234982449144438050} a^{15} + \frac{3612667744674584}{211010681020415656275} a^{14} - \frac{158759661260265388}{110529404344027248525} a^{13} + \frac{149564907846852427}{663176426064163491150} a^{12} - \frac{2897698530864257}{1039461482859190425} a^{11} + \frac{328970738856002152}{110529404344027248525} a^{10} - \frac{78434892224401333529}{2321117491224572219025} a^{9} - \frac{24402184945598640833}{464223498244914443805} a^{8} - \frac{31781388689369268529}{773705830408190739675} a^{7} - \frac{14288929335216116021}{331588213032081745575} a^{6} + \frac{5271729001057472516}{331588213032081745575} a^{5} + \frac{3582744191535143582}{7368626956268483235} a^{4} + \frac{209656349317520112109}{515803886938793826450} a^{3} - \frac{267881482019856202}{818736328474275915} a^{2} + \frac{1883908992494746484}{4093681642371379575} a + \frac{71829774715203709}{6367949221466590450}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{24}$, which has order $1536$ (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:  \( \frac{11534920205552603}{13263528521283269823} a^{15} - \frac{34002515113469863}{12057753201166608930} a^{14} + \frac{528270425023034596}{22105880868805449705} a^{13} - \frac{403684994501195092}{13263528521283269823} a^{12} + \frac{61965666319317194}{207892296571838085} a^{11} - \frac{6279425282397473164}{22105880868805449705} a^{10} + \frac{192866449298116281482}{66317642606416349115} a^{9} - \frac{17628948249965587747}{13263528521283269823} a^{8} + \frac{334256059142108560264}{22105880868805449705} a^{7} - \frac{1526131227462003227128}{66317642606416349115} a^{6} + \frac{625700314695054558778}{66317642606416349115} a^{5} - \frac{31640272318187983016}{1473725391253696647} a^{4} + \frac{30958488597743954033}{1473725391253696647} a^{3} + \frac{9694751995797215}{109164843796570122} a^{2} + \frac{5168238642752747542}{818736328474275915} a - \frac{616338840455100296}{90970703163808435} \) (order $6$)
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:  \( 1548459.9054883746 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-390}) \), \(\Q(\sqrt{195}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{130}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{78}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-2}, \sqrt{195})\), \(\Q(\sqrt{-3}, \sqrt{130})\), \(\Q(\sqrt{6}, \sqrt{-65})\), \(\Q(\sqrt{6}, \sqrt{130})\), \(\Q(\sqrt{-3}, \sqrt{-65})\), \(\Q(\sqrt{-2}, \sqrt{-65})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{10}, \sqrt{-39})\), \(\Q(\sqrt{-5}, \sqrt{78})\), \(\Q(\sqrt{13}, \sqrt{-30})\), \(\Q(\sqrt{15}, \sqrt{-26})\), \(\Q(\sqrt{10}, \sqrt{78})\), \(\Q(\sqrt{-5}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{15})\), \(\Q(\sqrt{-26}, \sqrt{-30})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{-39})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{10}, \sqrt{13})\), \(\Q(\sqrt{-30}, \sqrt{-39})\), \(\Q(\sqrt{15}, \sqrt{78})\), \(\Q(\sqrt{-5}, \sqrt{-26})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-26})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{6}, \sqrt{-26})\), \(\Q(\sqrt{6}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\sqrt{10}, \sqrt{-26})\), \(\Q(\sqrt{15}, \sqrt{-39})\), \(\Q(\sqrt{-30}, \sqrt{-65})\), \(\Q(\sqrt{-5}, \sqrt{13})\), 8.0.94758543360000.202, 8.0.94758543360000.147, 8.0.94758543360000.132, 8.0.5922408960000.16, 8.0.94758543360000.197, 8.0.94758543360000.161, 8.0.94758543360000.142, 8.8.94758543360000.10, 8.0.94758543360000.144, 8.0.94758543360000.101, 8.0.370150560000.4, 8.0.1169858560000.2, 8.0.94758543360000.111, 8.0.3317760000.8, 8.0.9475854336.1

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 R R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$