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Properties

Label 16.16.1373509658...9809.1
Degree $16$
Signature $[16, 0]$
Discriminant $13^{8}\cdot 17^{14}$
Root discriminant $43.01$
Ramified primes $13, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![101, -1001, 2122, 4745, -19654, 10286, 22809, -20610, -7614, 10181, 500, -2062, 145, 184, -24, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 24*x^14 + 184*x^13 + 145*x^12 - 2062*x^11 + 500*x^10 + 10181*x^9 - 7614*x^8 - 20610*x^7 + 22809*x^6 + 10286*x^5 - 19654*x^4 + 4745*x^3 + 2122*x^2 - 1001*x + 101)
 
gp: K = bnfinit(x^16 - 6*x^15 - 24*x^14 + 184*x^13 + 145*x^12 - 2062*x^11 + 500*x^10 + 10181*x^9 - 7614*x^8 - 20610*x^7 + 22809*x^6 + 10286*x^5 - 19654*x^4 + 4745*x^3 + 2122*x^2 - 1001*x + 101, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 24 x^{14} + 184 x^{13} + 145 x^{12} - 2062 x^{11} + 500 x^{10} + 10181 x^{9} - 7614 x^{8} - 20610 x^{7} + 22809 x^{6} + 10286 x^{5} - 19654 x^{4} + 4745 x^{3} + 2122 x^{2} - 1001 x + 101 \)

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:  \(137350965859713069141239809=13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.01$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(64,·)$, $\chi_{221}(1,·)$, $\chi_{221}(66,·)$, $\chi_{221}(196,·)$, $\chi_{221}(77,·)$, $\chi_{221}(144,·)$, $\chi_{221}(25,·)$, $\chi_{221}(155,·)$, $\chi_{221}(220,·)$, $\chi_{221}(157,·)$, $\chi_{221}(38,·)$, $\chi_{221}(103,·)$, $\chi_{221}(168,·)$, $\chi_{221}(53,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{76669531881690182200226} a^{15} + \frac{7885704384809375146069}{38334765940845091100113} a^{14} - \frac{35771192732010134199}{372182193600437777671} a^{13} - \frac{4176223152180388526899}{38334765940845091100113} a^{12} - \frac{5485540271429462743442}{38334765940845091100113} a^{11} + \frac{8784057657347067941560}{38334765940845091100113} a^{10} - \frac{4311530234552026075483}{38334765940845091100113} a^{9} + \frac{15461474667543901692414}{38334765940845091100113} a^{8} - \frac{161001171057625558165}{379552138028169218813} a^{7} + \frac{15445879708733862216531}{38334765940845091100113} a^{6} + \frac{6195750502396976116600}{38334765940845091100113} a^{5} - \frac{14810762571702438552790}{38334765940845091100113} a^{4} + \frac{1687296885786323412731}{38334765940845091100113} a^{3} - \frac{14334426265166406241522}{38334765940845091100113} a^{2} - \frac{7343996697667979125162}{38334765940845091100113} a - \frac{244284234354566567485}{759104276056338437626}$

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:  \( 71925461.2875 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{17})\), 4.4.4913.1, 4.4.830297.1, 8.8.689393108209.1, 8.8.11719682839553.1, \(\Q(\zeta_{17})^+\)

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$