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Properties

Label 16.8.90390672967...1729.1
Degree $16$
Signature $[8, 4]$
Discriminant $61^{4}\cdot 97^{14}$
Root discriminant $153.02$
Ramified primes $61, 97$
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![-171211, 602034, -512251, -496607, 1227748, -900440, 134579, 266588, -202626, 45433, 11235, -8165, 658, 286, -49, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 49*x^14 + 286*x^13 + 658*x^12 - 8165*x^11 + 11235*x^10 + 45433*x^9 - 202626*x^8 + 266588*x^7 + 134579*x^6 - 900440*x^5 + 1227748*x^4 - 496607*x^3 - 512251*x^2 + 602034*x - 171211)
 
gp: K = bnfinit(x^16 - 4*x^15 - 49*x^14 + 286*x^13 + 658*x^12 - 8165*x^11 + 11235*x^10 + 45433*x^9 - 202626*x^8 + 266588*x^7 + 134579*x^6 - 900440*x^5 + 1227748*x^4 - 496607*x^3 - 512251*x^2 + 602034*x - 171211, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 49 x^{14} + 286 x^{13} + 658 x^{12} - 8165 x^{11} + 11235 x^{10} + 45433 x^{9} - 202626 x^{8} + 266588 x^{7} + 134579 x^{6} - 900440 x^{5} + 1227748 x^{4} - 496607 x^{3} - 512251 x^{2} + 602034 x - 171211 \)

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:  \(90390672967514567934462041097091729=61^{4}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $153.02$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{61} a^{14} + \frac{15}{61} a^{13} - \frac{22}{61} a^{12} + \frac{24}{61} a^{11} + \frac{19}{61} a^{10} - \frac{27}{61} a^{9} + \frac{25}{61} a^{8} - \frac{13}{61} a^{7} + \frac{29}{61} a^{6} + \frac{19}{61} a^{5} + \frac{29}{61} a^{4} + \frac{22}{61} a^{3} + \frac{13}{61} a^{2} - \frac{6}{61} a - \frac{25}{61}$, $\frac{1}{229417744665592097121374867307424813834327} a^{15} + \frac{192972385543917925191585460459148222177}{229417744665592097121374867307424813834327} a^{14} + \frac{33720190921558995866251342895886756975883}{229417744665592097121374867307424813834327} a^{13} - \frac{105349537897937784104385066007110182779777}{229417744665592097121374867307424813834327} a^{12} - \frac{97554300286782169266759536300173022258376}{229417744665592097121374867307424813834327} a^{11} - \frac{100063625045535305640986844134785703059612}{229417744665592097121374867307424813834327} a^{10} - \frac{105892200508441031293122538571472923715657}{229417744665592097121374867307424813834327} a^{9} - \frac{2088899541464033797864045991648682396778}{229417744665592097121374867307424813834327} a^{8} - \frac{569381184136745816749846148772977504215}{3760946633862165526579915857498767439907} a^{7} + \frac{10634873806844341388078805791015063985788}{229417744665592097121374867307424813834327} a^{6} - \frac{7743117871028649493310644349515095506668}{229417744665592097121374867307424813834327} a^{5} + \frac{2112658356593773074208140474519271395696}{229417744665592097121374867307424813834327} a^{4} - \frac{26131380138545354635457797430844006655695}{229417744665592097121374867307424813834327} a^{3} - \frac{37321830045400056865634111795873634097690}{229417744665592097121374867307424813834327} a^{2} - \frac{21036946258484678969364283862405298882093}{229417744665592097121374867307424813834327} a + \frac{86673616069465510368635440577776042348017}{229417744665592097121374867307424813834327}$

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:  $11$
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:  \( 297876026262 \) (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

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
61Data not computed
$97$97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.1$x^{8} - 97$$8$$1$$7$$C_8$$[\ ]_{8}$