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Properties

Label 16.16.2475963176...0000.2
Degree $16$
Signature $[16, 0]$
Discriminant $2^{44}\cdot 5^{12}\cdot 7^{8}$
Root discriminant $59.51$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![191, -10884, 50918, -47356, -83845, 123300, 29964, -86152, 4831, 25128, -4076, -3348, 703, 196, -46, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 46*x^14 + 196*x^13 + 703*x^12 - 3348*x^11 - 4076*x^10 + 25128*x^9 + 4831*x^8 - 86152*x^7 + 29964*x^6 + 123300*x^5 - 83845*x^4 - 47356*x^3 + 50918*x^2 - 10884*x + 191)
 
gp: K = bnfinit(x^16 - 4*x^15 - 46*x^14 + 196*x^13 + 703*x^12 - 3348*x^11 - 4076*x^10 + 25128*x^9 + 4831*x^8 - 86152*x^7 + 29964*x^6 + 123300*x^5 - 83845*x^4 - 47356*x^3 + 50918*x^2 - 10884*x + 191, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 46 x^{14} + 196 x^{13} + 703 x^{12} - 3348 x^{11} - 4076 x^{10} + 25128 x^{9} + 4831 x^{8} - 86152 x^{7} + 29964 x^{6} + 123300 x^{5} - 83845 x^{4} - 47356 x^{3} + 50918 x^{2} - 10884 x + 191 \)

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:  \(24759631762948096000000000000=2^{44}\cdot 5^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.51$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(517,·)$, $\chi_{560}(449,·)$, $\chi_{560}(13,·)$, $\chi_{560}(141,·)$, $\chi_{560}(281,·)$, $\chi_{560}(153,·)$, $\chi_{560}(29,·)$, $\chi_{560}(421,·)$, $\chi_{560}(97,·)$, $\chi_{560}(293,·)$, $\chi_{560}(169,·)$, $\chi_{560}(237,·)$, $\chi_{560}(433,·)$, $\chi_{560}(309,·)$, $\chi_{560}(377,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{11} + \frac{2}{7} a^{9} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{21} a^{12} - \frac{1}{21} a^{10} - \frac{10}{21} a^{9} + \frac{5}{21} a^{8} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{4}{21} a^{5} + \frac{1}{21} a^{4} + \frac{3}{7} a^{3} + \frac{4}{21} a^{2} + \frac{1}{21} a + \frac{2}{21}$, $\frac{1}{21} a^{13} - \frac{1}{21} a^{11} - \frac{1}{21} a^{10} - \frac{1}{3} a^{9} - \frac{2}{7} a^{7} + \frac{2}{21} a^{6} - \frac{8}{21} a^{5} - \frac{2}{7} a^{4} + \frac{10}{21} a^{3} - \frac{8}{21} a^{2} + \frac{8}{21} a - \frac{2}{7}$, $\frac{1}{234832731} a^{14} + \frac{1060908}{78277577} a^{13} + \frac{2233094}{234832731} a^{12} - \frac{6645844}{234832731} a^{11} + \frac{3607826}{234832731} a^{10} - \frac{37774273}{78277577} a^{9} - \frac{10822748}{78277577} a^{8} - \frac{57840838}{234832731} a^{7} + \frac{22358107}{234832731} a^{6} - \frac{9244861}{78277577} a^{5} - \frac{11573924}{234832731} a^{4} - \frac{31757564}{234832731} a^{3} - \frac{65688124}{234832731} a^{2} + \frac{1308458}{78277577} a - \frac{37895665}{78277577}$, $\frac{1}{225180664035532989} a^{15} - \frac{81080919}{75060221345177663} a^{14} - \frac{1277423425350533}{225180664035532989} a^{13} + \frac{476930690893559}{32168666290790427} a^{12} - \frac{635705002277040}{75060221345177663} a^{11} - \frac{798324575490347}{32168666290790427} a^{10} + \frac{104062792060281559}{225180664035532989} a^{9} - \frac{33554395379553190}{225180664035532989} a^{8} + \frac{34685313679382911}{225180664035532989} a^{7} - \frac{72611924240876648}{225180664035532989} a^{6} - \frac{25712423651005233}{75060221345177663} a^{5} + \frac{53113557405302377}{225180664035532989} a^{4} - \frac{40832330400717236}{225180664035532989} a^{3} - \frac{98177726558363020}{225180664035532989} a^{2} + \frac{10921304167400323}{32168666290790427} a - \frac{84562542581543}{392985452068993}$

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:  \( 673805748.205718 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{16})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.51200.1, 4.4.6125.1, 4.4.392000.1, 4.4.12544000.2, 4.4.12544000.1, 8.8.2621440000.1, 8.8.153664000000.1, 8.8.157351936000000.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$