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Properties

Label 16.4.90390672967...1729.3
Degree $16$
Signature $[4, 6]$
Discriminant $61^{4}\cdot 97^{14}$
Root discriminant $153.02$
Ramified primes $61, 97$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group 16T1194

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1778071, -3390345, -4972170, 1432911, 1404898, 2210218, -1549932, 335929, 127136, -115640, 24126, -150, -1285, 295, 1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + x^14 + 295*x^13 - 1285*x^12 - 150*x^11 + 24126*x^10 - 115640*x^9 + 127136*x^8 + 335929*x^7 - 1549932*x^6 + 2210218*x^5 + 1404898*x^4 + 1432911*x^3 - 4972170*x^2 - 3390345*x - 1778071)
 
gp: K = bnfinit(x^16 - 3*x^15 + x^14 + 295*x^13 - 1285*x^12 - 150*x^11 + 24126*x^10 - 115640*x^9 + 127136*x^8 + 335929*x^7 - 1549932*x^6 + 2210218*x^5 + 1404898*x^4 + 1432911*x^3 - 4972170*x^2 - 3390345*x - 1778071, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + x^{14} + 295 x^{13} - 1285 x^{12} - 150 x^{11} + 24126 x^{10} - 115640 x^{9} + 127136 x^{8} + 335929 x^{7} - 1549932 x^{6} + 2210218 x^{5} + 1404898 x^{4} + 1432911 x^{3} - 4972170 x^{2} - 3390345 x - 1778071 \)

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:  \(90390672967514567934462041097091729=61^{4}\cdot 97^{14}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2398390816538597610541725647395234554353825536326625567} a^{15} - \frac{563146990457673344983957939612691954069723534070119132}{2398390816538597610541725647395234554353825536326625567} a^{14} + \frac{245361073992269772537631010623385867136167091849097600}{2398390816538597610541725647395234554353825536326625567} a^{13} + \frac{916986782049951904524811295556993915060541470880981414}{2398390816538597610541725647395234554353825536326625567} a^{12} + \frac{421837183905061442803579271744521191525630550528422791}{2398390816538597610541725647395234554353825536326625567} a^{11} - \frac{1124057021028135722235542685516977444277333806066698929}{2398390816538597610541725647395234554353825536326625567} a^{10} - \frac{306655409151423017235158695262633935144629359531071199}{2398390816538597610541725647395234554353825536326625567} a^{9} + \frac{802854889263760132310930870154660389244639933352963371}{2398390816538597610541725647395234554353825536326625567} a^{8} - \frac{725378488225729821546730759945799127436618147226079496}{2398390816538597610541725647395234554353825536326625567} a^{7} - \frac{285498926144122723175363532829699118921643390077043055}{2398390816538597610541725647395234554353825536326625567} a^{6} + \frac{442042632001077067501101040159364648263558274357273377}{2398390816538597610541725647395234554353825536326625567} a^{5} - \frac{365427918942489737771697665005102511933669347090881453}{2398390816538597610541725647395234554353825536326625567} a^{4} + \frac{95407674807105200858905545373185382124273892760101710}{2398390816538597610541725647395234554353825536326625567} a^{3} - \frac{1150284572027797061890844575314991203527863619828597283}{2398390816538597610541725647395234554353825536326625567} a^{2} + \frac{875065362497932900730882069216281738698438634876346323}{2398390816538597610541725647395234554353825536326625567} a + \frac{525870295497885834528039930992596008001510922349644962}{2398390816538597610541725647395234554353825536326625567}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 19039859663.9 \) (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}$