↑

Properties

Label 16.12.9039067296...1729.1
Degree $16$
Signature $[12, 2]$
Discriminant $61^{4}\cdot 97^{14}$
Root discriminant $153.02$
Ramified primes $61, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T565)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11903533, 4282958, 24199071, -7154965, -16209686, 3143799, 4598000, -241659, -548175, -76067, 20561, 8150, 481, -122, -48, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 48*x^14 - 122*x^13 + 481*x^12 + 8150*x^11 + 20561*x^10 - 76067*x^9 - 548175*x^8 - 241659*x^7 + 4598000*x^6 + 3143799*x^5 - 16209686*x^4 - 7154965*x^3 + 24199071*x^2 + 4282958*x - 11903533)
 
gp: K = bnfinit(x^16 - 3*x^15 - 48*x^14 - 122*x^13 + 481*x^12 + 8150*x^11 + 20561*x^10 - 76067*x^9 - 548175*x^8 - 241659*x^7 + 4598000*x^6 + 3143799*x^5 - 16209686*x^4 - 7154965*x^3 + 24199071*x^2 + 4282958*x - 11903533, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 48 x^{14} - 122 x^{13} + 481 x^{12} + 8150 x^{11} + 20561 x^{10} - 76067 x^{9} - 548175 x^{8} - 241659 x^{7} + 4598000 x^{6} + 3143799 x^{5} - 16209686 x^{4} - 7154965 x^{3} + 24199071 x^{2} + 4282958 x - 11903533 \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{452} a^{14} + \frac{5}{113} a^{13} - \frac{14}{113} a^{12} + \frac{39}{226} a^{11} - \frac{53}{226} a^{10} - \frac{14}{113} a^{9} - \frac{49}{452} a^{8} + \frac{45}{226} a^{7} + \frac{17}{452} a^{6} - \frac{105}{226} a^{5} + \frac{13}{452} a^{4} - \frac{11}{113} a^{3} - \frac{39}{452} a^{2} - \frac{56}{113} a$, $\frac{1}{20333800176097657649863671229936634362773770314852} a^{15} - \frac{4176080700767817746987503309392374332316334035}{10166900088048828824931835614968317181386885157426} a^{14} - \frac{581209206782096714856951566334391467288440173163}{5083450044024414412465917807484158590693442578713} a^{13} - \frac{2000272160976763911622695044924080294378106554949}{20333800176097657649863671229936634362773770314852} a^{12} - \frac{3897197475516565597288431695470693521717514973}{18897583806782209711769211180238507772094582077} a^{11} + \frac{286179202473006978460446468926651300215500160261}{5083450044024414412465917807484158590693442578713} a^{10} + \frac{1923334115196126581817438625328365637718041607735}{20333800176097657649863671229936634362773770314852} a^{9} + \frac{902048954841425459836443896544976592363716529185}{5083450044024414412465917807484158590693442578713} a^{8} - \frac{3874699394083622003183669449230228672501635370401}{20333800176097657649863671229936634362773770314852} a^{7} + \frac{1899594196456594597414901958206276755617826343391}{20333800176097657649863671229936634362773770314852} a^{6} - \frac{8680633780326329030778405241178722930410924036361}{20333800176097657649863671229936634362773770314852} a^{5} - \frac{3668270550523329786547796180103811123512958405803}{20333800176097657649863671229936634362773770314852} a^{4} - \frac{9948991006314122589068077120744748863951682196587}{20333800176097657649863671229936634362773770314852} a^{3} - \frac{9362829643946160351192762842773211727368348372181}{20333800176097657649863671229936634362773770314852} a^{2} + \frac{2054941220026006544520966234039765573181959280934}{5083450044024414412465917807484158590693442578713} a + \frac{46599338616895715661069662426889979243748497917}{179945134301749182742156382565810923564369648804}$

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:  \( 800848035390 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T565):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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}$