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Properties

Label 16.12.1752146415...2529.1
Degree $16$
Signature $[12, 2]$
Discriminant $17^{14}\cdot 101^{4}$
Root discriminant $37.82$
Ramified primes $17, 101$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-239, -2572, -1182, 15687, 10486, -26215, -11064, 12647, 776, -1502, 1179, -46, -182, 23, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 4*x^14 + 23*x^13 - 182*x^12 - 46*x^11 + 1179*x^10 - 1502*x^9 + 776*x^8 + 12647*x^7 - 11064*x^6 - 26215*x^5 + 10486*x^4 + 15687*x^3 - 1182*x^2 - 2572*x - 239)
 
gp: K = bnfinit(x^16 - 2*x^15 - 4*x^14 + 23*x^13 - 182*x^12 - 46*x^11 + 1179*x^10 - 1502*x^9 + 776*x^8 + 12647*x^7 - 11064*x^6 - 26215*x^5 + 10486*x^4 + 15687*x^3 - 1182*x^2 - 2572*x - 239, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 4 x^{14} + 23 x^{13} - 182 x^{12} - 46 x^{11} + 1179 x^{10} - 1502 x^{9} + 776 x^{8} + 12647 x^{7} - 11064 x^{6} - 26215 x^{5} + 10486 x^{4} + 15687 x^{3} - 1182 x^{2} - 2572 x - 239 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17521464151279710991512529=17^{14}\cdot 101^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.82$
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:  $17, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{239} a^{14} - \frac{34}{239} a^{13} + \frac{111}{239} a^{12} - \frac{83}{239} a^{11} + \frac{109}{239} a^{10} + \frac{28}{239} a^{9} + \frac{103}{239} a^{8} - \frac{16}{239} a^{7} + \frac{15}{239} a^{6} + \frac{11}{239} a^{5} + \frac{40}{239} a^{4} + \frac{42}{239} a^{3} + \frac{97}{239} a^{2} - \frac{81}{239} a$, $\frac{1}{1956502386941757401809357720757} a^{15} + \frac{1155020458287307086852401662}{1956502386941757401809357720757} a^{14} - \frac{3400523611728348750470993640}{19371310761799578235736215057} a^{13} + \frac{791907489645686933395395538938}{1956502386941757401809357720757} a^{12} + \frac{381889154147929386521919884606}{1956502386941757401809357720757} a^{11} - \frac{767301973977301140761726126456}{1956502386941757401809357720757} a^{10} - \frac{418165506450071012988363528678}{1956502386941757401809357720757} a^{9} - \frac{842029109041645576985575802394}{1956502386941757401809357720757} a^{8} - \frac{435363037999789149547613837132}{1956502386941757401809357720757} a^{7} + \frac{134594000631029352860881516172}{1956502386941757401809357720757} a^{6} + \frac{505610506317948027269102836415}{1956502386941757401809357720757} a^{5} + \frac{951932116280779057725954725053}{1956502386941757401809357720757} a^{4} + \frac{21201934161547241779565439141}{1956502386941757401809357720757} a^{3} + \frac{752955901748882742290311074152}{1956502386941757401809357720757} a^{2} - \frac{354600802414593585232939072377}{1956502386941757401809357720757} a - \frac{1219531733307688964671303837}{8186202455823252727235806363}$

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:  $13$
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:  \( 7947040.59106 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1194:

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 t16n1194
Character table for t16n1194 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$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}$
101Data not computed