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Properties

Label 16.0.59710443940...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 13^{12}$
Root discriminant $26.52$
Ramified primes $3, 5, 13$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -108, 126, -384, -89, 1090, 632, 599, 1014, 452, 163, 184, 6, -41, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 4*x^14 - 41*x^13 + 6*x^12 + 184*x^11 + 163*x^10 + 452*x^9 + 1014*x^8 + 599*x^7 + 632*x^6 + 1090*x^5 - 89*x^4 - 384*x^3 + 126*x^2 - 108*x + 81)
 
gp: K = bnfinit(x^16 - x^15 - 4*x^14 - 41*x^13 + 6*x^12 + 184*x^11 + 163*x^10 + 452*x^9 + 1014*x^8 + 599*x^7 + 632*x^6 + 1090*x^5 - 89*x^4 - 384*x^3 + 126*x^2 - 108*x + 81, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 4 x^{14} - 41 x^{13} + 6 x^{12} + 184 x^{11} + 163 x^{10} + 452 x^{9} + 1014 x^{8} + 599 x^{7} + 632 x^{6} + 1090 x^{5} - 89 x^{4} - 384 x^{3} + 126 x^{2} - 108 x + 81 \)

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:  \(59710443940858531640625=3^{8}\cdot 5^{8}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.52$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(64,·)$, $\chi_{195}(1,·)$, $\chi_{195}(194,·)$, $\chi_{195}(131,·)$, $\chi_{195}(14,·)$, $\chi_{195}(79,·)$, $\chi_{195}(86,·)$, $\chi_{195}(151,·)$, $\chi_{195}(31,·)$, $\chi_{195}(161,·)$, $\chi_{195}(34,·)$, $\chi_{195}(164,·)$, $\chi_{195}(44,·)$, $\chi_{195}(109,·)$, $\chi_{195}(116,·)$, $\chi_{195}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{3}$, $\frac{1}{1098} a^{12} - \frac{13}{549} a^{11} + \frac{19}{549} a^{10} + \frac{7}{366} a^{9} - \frac{86}{549} a^{8} + \frac{253}{549} a^{7} + \frac{127}{1098} a^{6} + \frac{8}{183} a^{5} + \frac{230}{549} a^{4} - \frac{293}{1098} a^{3} + \frac{16}{549} a^{2} - \frac{19}{61} a + \frac{3}{122}$, $\frac{1}{25254} a^{13} + \frac{1}{25254} a^{12} + \frac{583}{12627} a^{11} - \frac{505}{8418} a^{10} - \frac{2167}{25254} a^{9} + \frac{2872}{12627} a^{8} - \frac{4511}{25254} a^{7} - \frac{1}{2} a^{6} - \frac{1135}{12627} a^{5} - \frac{5441}{25254} a^{4} + \frac{11519}{25254} a^{3} - \frac{157}{4209} a^{2} - \frac{1361}{8418} a + \frac{1301}{2806}$, $\frac{1}{25254} a^{14} - \frac{4}{12627} a^{12} + \frac{2563}{25254} a^{11} - \frac{1568}{12627} a^{10} + \frac{19}{4209} a^{9} - \frac{10531}{25254} a^{8} + \frac{2221}{12627} a^{7} - \frac{1963}{12627} a^{6} - \frac{1}{46} a^{5} + \frac{3857}{12627} a^{4} + \frac{1463}{12627} a^{3} + \frac{157}{2806} a^{2} + \frac{746}{4209} a + \frac{396}{1403}$, $\frac{1}{1415086651386} a^{15} + \frac{4034207}{1415086651386} a^{14} - \frac{11157703}{1415086651386} a^{13} - \frac{155406862}{707543325693} a^{12} + \frac{27897899549}{471695550462} a^{11} - \frac{5291548411}{61525506582} a^{10} - \frac{15329887450}{707543325693} a^{9} + \frac{302755149719}{1415086651386} a^{8} + \frac{1111462519}{27746797086} a^{7} + \frac{129327599809}{707543325693} a^{6} - \frac{638638140385}{1415086651386} a^{5} - \frac{136294113317}{1415086651386} a^{4} + \frac{26200061204}{707543325693} a^{3} + \frac{20043153485}{52410616718} a^{2} - \frac{67587586199}{157231850154} a - \frac{7478951279}{52410616718}$

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

Class group and class number

$C_{4}$, 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:  \( \frac{2813694739}{1415086651386} a^{15} + \frac{779000785}{707543325693} a^{14} - \frac{10063753958}{707543325693} a^{13} - \frac{60388869250}{707543325693} a^{12} - \frac{8816084557}{78615925077} a^{11} + \frac{356818299035}{707543325693} a^{10} + \frac{7644541961}{11599070913} a^{9} + \frac{735238981588}{707543325693} a^{8} + \frac{50053067578}{13873398543} a^{7} + \frac{2290213078519}{707543325693} a^{6} + \frac{1546986988594}{707543325693} a^{5} + \frac{3952159141403}{707543325693} a^{4} + \frac{1458168790721}{707543325693} a^{3} - \frac{287791191752}{235847775231} a^{2} + \frac{123128704825}{78615925077} a - \frac{3953229047}{52410616718} \) (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:  \( 57289.5984617 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-195}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{65})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-15}, \sqrt{-39})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{13}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-39})\), 4.4.494325.1, 4.4.19773.1, 4.0.54925.1, 4.0.2197.1, 8.0.1445900625.1, 8.8.244357205625.1, 8.0.3016755625.1, 8.0.244357205625.3, 8.0.390971529.1, 8.0.244357205625.1, 8.0.244357205625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$