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Properties

Label 16.0.79253796088...3601.6
Degree $16$
Signature $[0, 8]$
Discriminant $37^{8}\cdot 41^{12}$
Root discriminant $98.56$
Ramified primes $37, 41$
Class number $96$ (GRH)
Class group $[2, 48]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![114193360, -8995382, 12528339, -5387882, -1294141, 4221358, -511450, -680600, 67167, 59738, -3526, -3980, 178, 196, -8, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 8*x^14 + 196*x^13 + 178*x^12 - 3980*x^11 - 3526*x^10 + 59738*x^9 + 67167*x^8 - 680600*x^7 - 511450*x^6 + 4221358*x^5 - 1294141*x^4 - 5387882*x^3 + 12528339*x^2 - 8995382*x + 114193360)
 
gp: K = bnfinit(x^16 - 8*x^15 - 8*x^14 + 196*x^13 + 178*x^12 - 3980*x^11 - 3526*x^10 + 59738*x^9 + 67167*x^8 - 680600*x^7 - 511450*x^6 + 4221358*x^5 - 1294141*x^4 - 5387882*x^3 + 12528339*x^2 - 8995382*x + 114193360, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 8 x^{14} + 196 x^{13} + 178 x^{12} - 3980 x^{11} - 3526 x^{10} + 59738 x^{9} + 67167 x^{8} - 680600 x^{7} - 511450 x^{6} + 4221358 x^{5} - 1294141 x^{4} - 5387882 x^{3} + 12528339 x^{2} - 8995382 x + 114193360 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(79253796088782001552124351073601=37^{8}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.56$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{30} a^{11} + \frac{1}{15} a^{10} + \frac{1}{15} a^{9} + \frac{1}{30} a^{8} - \frac{1}{2} a^{7} - \frac{7}{15} a^{6} - \frac{1}{10} a^{5} + \frac{7}{15} a^{4} - \frac{1}{5} a^{3} + \frac{7}{30} a^{2} + \frac{11}{30} a - \frac{1}{3}$, $\frac{1}{30} a^{12} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{2}{15} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} + \frac{1}{30} a^{4} + \frac{7}{15} a^{3} - \frac{1}{10} a^{2} + \frac{13}{30} a$, $\frac{1}{30} a^{13} + \frac{1}{30} a^{10} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{1}{2} a^{7} - \frac{4}{15} a^{6} - \frac{1}{6} a^{5} - \frac{1}{10} a^{4} - \frac{13}{30} a^{2} - \frac{4}{15} a$, $\frac{1}{31048572455651935815270} a^{14} - \frac{7}{31048572455651935815270} a^{13} - \frac{282481466301394063877}{31048572455651935815270} a^{12} - \frac{25001068837895422511}{2069904830376795721018} a^{11} + \frac{1217762163602840415746}{15524286227825967907635} a^{10} - \frac{402669743853366649231}{15524286227825967907635} a^{9} - \frac{65517411110645539758}{5174762075941989302545} a^{8} + \frac{455855686928629606187}{6209714491130387163054} a^{7} - \frac{2484136399207687256939}{6209714491130387163054} a^{6} - \frac{81882730796057318739}{2069904830376795721018} a^{5} - \frac{942523568704329155416}{3104857245565193581527} a^{4} - \frac{330965230758475220}{29017357435188725061} a^{3} - \frac{11710167986309253085543}{31048572455651935815270} a^{2} + \frac{5790274907933527528417}{15524286227825967907635} a + \frac{195597801835107847368}{1034952415188397860509}$, $\frac{1}{16838106571286879571658150050} a^{15} + \frac{5423}{336762131425737591433163001} a^{14} + \frac{8808338764935303492811111}{8419053285643439785829075025} a^{13} - \frac{63701790714259735431536541}{5612702190428959857219383350} a^{12} + \frac{9548132445450332484292457}{8419053285643439785829075025} a^{11} + \frac{418562473352869943303280619}{5612702190428959857219383350} a^{10} + \frac{286618960831201333186253}{1122540438085791971443876670} a^{9} + \frac{318822278128837586737351101}{5612702190428959857219383350} a^{8} - \frac{6984183070325799968642947639}{16838106571286879571658150050} a^{7} - \frac{14478991425428160975371888}{78682740987321867157281075} a^{6} - \frac{564295144614711944644652596}{2806351095214479928609691675} a^{5} - \frac{31568211307920075572830373}{673524262851475182866326002} a^{4} - \frac{2782857994562223124635631277}{5612702190428959857219383350} a^{3} - \frac{473763184258018872134308869}{1122540438085791971443876670} a^{2} - \frac{1937167781127638079310472197}{5612702190428959857219383350} a - \frac{132225965418926982026989172}{1683810657128687957165815005}$

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

Class group and class number

$C_{2}\times C_{48}$, which has order $96$ (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:  $7$
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:  \( 994157972.902 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.94352849.1, 4.0.62197.1, 4.0.2550077.1, 8.0.158607139169.1, 8.0.217133173522361.1, 8.0.8902460114416801.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$37$37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$