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Properties

Label 16.4.57681033264...3953.5
Degree $16$
Signature $[4, 6]$
Discriminant $17^{15}\cdot 67^{4}$
Root discriminant $40.74$
Ramified primes $17, 67$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group 16T1223

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32743, -39480, 48758, -14853, 29839, -5770, -655, -3584, -1905, 397, -36, 199, -38, 7, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 9*x^14 + 7*x^13 - 38*x^12 + 199*x^11 - 36*x^10 + 397*x^9 - 1905*x^8 - 3584*x^7 - 655*x^6 - 5770*x^5 + 29839*x^4 - 14853*x^3 + 48758*x^2 - 39480*x + 32743)
 
gp: K = bnfinit(x^16 - 3*x^15 + 9*x^14 + 7*x^13 - 38*x^12 + 199*x^11 - 36*x^10 + 397*x^9 - 1905*x^8 - 3584*x^7 - 655*x^6 - 5770*x^5 + 29839*x^4 - 14853*x^3 + 48758*x^2 - 39480*x + 32743, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 9 x^{14} + 7 x^{13} - 38 x^{12} + 199 x^{11} - 36 x^{10} + 397 x^{9} - 1905 x^{8} - 3584 x^{7} - 655 x^{6} - 5770 x^{5} + 29839 x^{4} - 14853 x^{3} + 48758 x^{2} - 39480 x + 32743 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57681033264163530732453953=17^{15}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.74$
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:  $17, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3}$, $\frac{1}{1294} a^{14} - \frac{235}{1294} a^{13} - \frac{199}{1294} a^{12} + \frac{126}{647} a^{11} + \frac{148}{647} a^{10} + \frac{105}{1294} a^{9} - \frac{359}{1294} a^{8} - \frac{211}{1294} a^{7} + \frac{183}{647} a^{6} - \frac{276}{647} a^{5} - \frac{25}{1294} a^{4} - \frac{485}{1294} a^{3} + \frac{461}{1294} a^{2} - \frac{132}{647} a - \frac{237}{647}$, $\frac{1}{271659073565758143159242453760014} a^{15} - \frac{43812483073157900456721393810}{135829536782879071579621226880007} a^{14} + \frac{33310179695584340420765230260415}{271659073565758143159242453760014} a^{13} + \frac{18751755758230567173937748174407}{135829536782879071579621226880007} a^{12} + \frac{14625610384882619334054490378687}{271659073565758143159242453760014} a^{11} - \frac{21188094990002340263466820380882}{135829536782879071579621226880007} a^{10} - \frac{56337658176240558197959368504448}{135829536782879071579621226880007} a^{9} - \frac{50559697941276568356289221336327}{271659073565758143159242453760014} a^{8} + \frac{21683697013202316775937675360415}{135829536782879071579621226880007} a^{7} + \frac{103314085438023763998378032214371}{271659073565758143159242453760014} a^{6} - \frac{51853903399227655027129298791129}{135829536782879071579621226880007} a^{5} - \frac{4097573336455632517748789741607}{135829536782879071579621226880007} a^{4} - \frac{36727289467220706118044329545547}{271659073565758143159242453760014} a^{3} - \frac{63726343660155232657623866972203}{135829536782879071579621226880007} a^{2} - \frac{104765347799977010315805040288553}{271659073565758143159242453760014} a + \frac{110905509041209733806953453203201}{271659073565758143159242453760014}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  $9$
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:  \( 431395.176696 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1223:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 40 conjugacy class representatives for t16n1223
Character table for t16n1223 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 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
17Data not computed
67Data not computed