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Properties

Label 16.0.19757093783...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 3^{12}\cdot 5^{12}\cdot 29^{6}$
Root discriminant $38.11$
Ramified primes $2, 3, 5, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^3.C_2^4.C_2$ (as 16T456)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17551, -33489, 31750, -73992, 129040, -94281, 11356, 23001, -11819, -489, 2188, -651, -62, 84, -14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 14*x^14 + 84*x^13 - 62*x^12 - 651*x^11 + 2188*x^10 - 489*x^9 - 11819*x^8 + 23001*x^7 + 11356*x^6 - 94281*x^5 + 129040*x^4 - 73992*x^3 + 31750*x^2 - 33489*x + 17551)
 
gp: K = bnfinit(x^16 - 3*x^15 - 14*x^14 + 84*x^13 - 62*x^12 - 651*x^11 + 2188*x^10 - 489*x^9 - 11819*x^8 + 23001*x^7 + 11356*x^6 - 94281*x^5 + 129040*x^4 - 73992*x^3 + 31750*x^2 - 33489*x + 17551, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 14 x^{14} + 84 x^{13} - 62 x^{12} - 651 x^{11} + 2188 x^{10} - 489 x^{9} - 11819 x^{8} + 23001 x^{7} + 11356 x^{6} - 94281 x^{5} + 129040 x^{4} - 73992 x^{3} + 31750 x^{2} - 33489 x + 17551 \)

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:  \(19757093783472562500000000=2^{8}\cdot 3^{12}\cdot 5^{12}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.11$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{12} a^{10} + \frac{1}{4} a^{7} + \frac{5}{12} a^{6} - \frac{1}{4} a^{5} + \frac{5}{12} a^{4} - \frac{1}{4} a^{3} - \frac{5}{12}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{8} + \frac{5}{12} a^{7} + \frac{1}{12} a^{6} + \frac{5}{12} a^{5} - \frac{1}{4} a^{4} + \frac{1}{3} a^{2} - \frac{5}{12} a - \frac{1}{3}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{12} a^{8} + \frac{5}{24} a^{7} - \frac{5}{24} a^{6} + \frac{7}{24} a^{5} + \frac{5}{12} a^{4} + \frac{7}{24} a^{3} - \frac{1}{24} a^{2} - \frac{11}{24} a + \frac{1}{24}$, $\frac{1}{24} a^{13} + \frac{1}{24} a^{9} + \frac{1}{24} a^{8} - \frac{1}{2} a^{7} - \frac{5}{12} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{5}{24}$, $\frac{1}{48} a^{14} - \frac{1}{48} a^{12} + \frac{1}{48} a^{11} + \frac{1}{24} a^{9} + \frac{1}{24} a^{8} + \frac{5}{16} a^{7} + \frac{1}{12} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{7}{48} a^{3} - \frac{1}{16} a^{2} - \frac{1}{8} a + \frac{7}{16}$, $\frac{1}{1562391304805422366984368} a^{15} - \frac{758772624949156317689}{390597826201355591746092} a^{14} - \frac{5979405363258248147629}{1562391304805422366984368} a^{13} + \frac{16442026579362993929021}{1562391304805422366984368} a^{12} - \frac{1297311073710873943163}{32549818850112965978841} a^{11} + \frac{9682135326816843063799}{260398550800903727830728} a^{10} - \frac{15128216868818882285041}{781195652402711183492184} a^{9} + \frac{6274412023987777220599}{1562391304805422366984368} a^{8} + \frac{39929081546576458559171}{390597826201355591746092} a^{7} - \frac{138052695695785676385469}{390597826201355591746092} a^{6} + \frac{26406878707275335793373}{65099637700225931957682} a^{5} - \frac{148591054102284829876499}{520797101601807455661456} a^{4} + \frac{1983880706642525513945}{91905370870907198057904} a^{3} - \frac{197755762088647515722845}{781195652402711183492184} a^{2} + \frac{69330664508809837177217}{1562391304805422366984368} a + \frac{28897146704205776416037}{97649456550338897936523}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( \frac{927545834201851}{384558432163830672} a^{15} - \frac{280767646156505}{64093072027305112} a^{14} - \frac{4985834959811495}{128186144054610224} a^{13} + \frac{20071806030706935}{128186144054610224} a^{12} + \frac{2230097249796205}{64093072027305112} a^{11} - \frac{97890416407234315}{64093072027305112} a^{10} + \frac{334193437772713435}{96139608040957668} a^{9} + \frac{372562108430059165}{128186144054610224} a^{8} - \frac{1605690612241429965}{64093072027305112} a^{7} + \frac{833100100484514825}{32046536013652556} a^{6} + \frac{3705483277245709021}{64093072027305112} a^{5} - \frac{20422928642883462755}{128186144054610224} a^{4} + \frac{2812176250161332765}{22621084244931216} a^{3} - \frac{1023864438735124315}{32046536013652556} a^{2} + \frac{4743985054559835165}{128186144054610224} a - \frac{1145906098265005029}{32046536013652556} \) (order $6$)
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:  \( 1115822.93458 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_2^4.C_2$ (as 16T456):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$
Character table for $C_2^3.C_2^4.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), 4.0.3625.1, 4.4.32625.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.1064390625.2

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$