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Properties

Label 16.4.33930019567...1409.2
Degree $16$
Signature $[4, 6]$
Discriminant $17^{14}\cdot 67^{4}$
Root discriminant $34.13$
Ramified primes $17, 67$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1021, 2145, -2560, 7472, -7926, 9313, -5195, 3402, -657, -773, 503, -199, 30, -19, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 10*x^14 - 19*x^13 + 30*x^12 - 199*x^11 + 503*x^10 - 773*x^9 - 657*x^8 + 3402*x^7 - 5195*x^6 + 9313*x^5 - 7926*x^4 + 7472*x^3 - 2560*x^2 + 2145*x + 1021)
 
gp: K = bnfinit(x^16 - 5*x^15 + 10*x^14 - 19*x^13 + 30*x^12 - 199*x^11 + 503*x^10 - 773*x^9 - 657*x^8 + 3402*x^7 - 5195*x^6 + 9313*x^5 - 7926*x^4 + 7472*x^3 - 2560*x^2 + 2145*x + 1021, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 10 x^{14} - 19 x^{13} + 30 x^{12} - 199 x^{11} + 503 x^{10} - 773 x^{9} - 657 x^{8} + 3402 x^{7} - 5195 x^{6} + 9313 x^{5} - 7926 x^{4} + 7472 x^{3} - 2560 x^{2} + 2145 x + 1021 \)

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:  \(3393001956715501807791409=17^{14}\cdot 67^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{13} - \frac{4}{13} a^{12} + \frac{1}{13} a^{11} - \frac{4}{13} a^{10} - \frac{4}{13} a^{9} - \frac{6}{13} a^{8} - \frac{3}{13} a^{7} + \frac{2}{13} a^{6} - \frac{5}{13} a^{5} + \frac{4}{13} a^{4} + \frac{6}{13} a^{3} + \frac{1}{13} a^{2} + \frac{6}{13} a + \frac{2}{13}$, $\frac{1}{18464745612758539507893776698987} a^{15} + \frac{648290566238181769026597567018}{18464745612758539507893776698987} a^{14} + \frac{4235450101793973823328784526485}{18464745612758539507893776698987} a^{13} - \frac{3333562884061217229079575592179}{18464745612758539507893776698987} a^{12} - \frac{8491786087652573640605412028301}{18464745612758539507893776698987} a^{11} - \frac{4384131034781048180065693562581}{18464745612758539507893776698987} a^{10} + \frac{1177480785383230808256997371003}{18464745612758539507893776698987} a^{9} - \frac{8541685735275388111613703149938}{18464745612758539507893776698987} a^{8} - \frac{1746521379308146148010386690315}{18464745612758539507893776698987} a^{7} - \frac{5824089599936383622947350311060}{18464745612758539507893776698987} a^{6} - \frac{6433028937882614656243830943116}{18464745612758539507893776698987} a^{5} - \frac{7387342852949820912378839883084}{18464745612758539507893776698987} a^{4} - \frac{251446193303792571133919484934}{18464745612758539507893776698987} a^{3} - \frac{1385171215612730081480154762244}{18464745612758539507893776698987} a^{2} + \frac{8186219482711401632986927424691}{18464745612758539507893776698987} a - \frac{6425811194658825708493597958460}{18464745612758539507893776698987}$

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:  $9$
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:  \( 295231.718761 \) (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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
67Data not computed