↑

Properties

Label 16.8.13816369346...4537.1
Degree $16$
Signature $[8, 4]$
Discriminant $13^{6}\cdot 17^{15}$
Root discriminant $37.26$
Ramified primes $13, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8329, 42102, -67305, 17830, 49797, -50365, 20885, 328, -7651, 7096, -3211, 887, -225, 28, 8, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 8*x^14 + 28*x^13 - 225*x^12 + 887*x^11 - 3211*x^10 + 7096*x^9 - 7651*x^8 + 328*x^7 + 20885*x^6 - 50365*x^5 + 49797*x^4 + 17830*x^3 - 67305*x^2 + 42102*x - 8329)
 
gp: K = bnfinit(x^16 - 5*x^15 + 8*x^14 + 28*x^13 - 225*x^12 + 887*x^11 - 3211*x^10 + 7096*x^9 - 7651*x^8 + 328*x^7 + 20885*x^6 - 50365*x^5 + 49797*x^4 + 17830*x^3 - 67305*x^2 + 42102*x - 8329, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 8 x^{14} + 28 x^{13} - 225 x^{12} + 887 x^{11} - 3211 x^{10} + 7096 x^{9} - 7651 x^{8} + 328 x^{7} + 20885 x^{6} - 50365 x^{5} + 49797 x^{4} + 17830 x^{3} - 67305 x^{2} + 42102 x - 8329 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13816369346835042457994537=13^{6}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.26$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{543314590645753137631226165730673207} a^{15} - \frac{216139157526902713867327888543318946}{543314590645753137631226165730673207} a^{14} - \frac{151615090825233108971244943982434887}{543314590645753137631226165730673207} a^{13} - \frac{186100108089370202659317413828897543}{543314590645753137631226165730673207} a^{12} + \frac{45280944971126493743379825618173522}{543314590645753137631226165730673207} a^{11} - \frac{168514240757323491428548307015263191}{543314590645753137631226165730673207} a^{10} + \frac{203884629910988986104566793490495880}{543314590645753137631226165730673207} a^{9} - \frac{49092603489289842375044942456075752}{543314590645753137631226165730673207} a^{8} + \frac{86124815120591883223431566108789650}{543314590645753137631226165730673207} a^{7} - \frac{109497263434833906778251818133464770}{543314590645753137631226165730673207} a^{6} + \frac{125327665453126332444933535410326357}{543314590645753137631226165730673207} a^{5} + \frac{28268796546147656098159331733584975}{543314590645753137631226165730673207} a^{4} + \frac{218452890766915078880252840722437113}{543314590645753137631226165730673207} a^{3} - \frac{251128208759084882104234916459046060}{543314590645753137631226165730673207} a^{2} - \frac{219517496461242963631129545180991118}{543314590645753137631226165730673207} a - \frac{132702671279905582863452605490817298}{543314590645753137631226165730673207}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 1478264.70026 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
17Data not computed