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Properties

Label 16.0.47937040118...2576.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 17^{8}$
Root discriminant $19.61$
Ramified primes $2, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2\times D_8$ (as 16T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![145, -884, 2362, -3744, 3606, -1328, -970, 804, 606, -1004, 614, -320, 158, -48, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 48*x^13 + 158*x^12 - 320*x^11 + 614*x^10 - 1004*x^9 + 606*x^8 + 804*x^7 - 970*x^6 - 1328*x^5 + 3606*x^4 - 3744*x^3 + 2362*x^2 - 884*x + 145)
 
gp: K = bnfinit(x^16 - 4*x^15 + 10*x^14 - 48*x^13 + 158*x^12 - 320*x^11 + 614*x^10 - 1004*x^9 + 606*x^8 + 804*x^7 - 970*x^6 - 1328*x^5 + 3606*x^4 - 3744*x^3 + 2362*x^2 - 884*x + 145, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 10 x^{14} - 48 x^{13} + 158 x^{12} - 320 x^{11} + 614 x^{10} - 1004 x^{9} + 606 x^{8} + 804 x^{7} - 970 x^{6} - 1328 x^{5} + 3606 x^{4} - 3744 x^{3} + 2362 x^{2} - 884 x + 145 \)

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:  \(479370401182778392576=2^{36}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.61$
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:  $2, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{3} - \frac{3}{10} a^{2} + \frac{1}{10} a$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{10} a^{3} + \frac{2}{5} a^{2} - \frac{1}{10} a$, $\frac{1}{50} a^{12} + \frac{1}{50} a^{11} - \frac{1}{25} a^{10} + \frac{1}{10} a^{9} + \frac{3}{25} a^{8} + \frac{12}{25} a^{7} + \frac{7}{25} a^{6} + \frac{4}{25} a^{5} + \frac{7}{50} a^{4} - \frac{11}{50} a^{3} + \frac{11}{25} a^{2} + \frac{1}{50} a + \frac{2}{5}$, $\frac{1}{50} a^{13} + \frac{1}{25} a^{11} + \frac{1}{25} a^{10} + \frac{1}{50} a^{9} - \frac{7}{50} a^{8} + \frac{2}{5} a^{7} + \frac{2}{25} a^{6} - \frac{1}{50} a^{5} + \frac{11}{25} a^{4} - \frac{6}{25} a^{3} + \frac{7}{25} a^{2} + \frac{9}{50} a + \frac{1}{10}$, $\frac{1}{250} a^{14} + \frac{1}{250} a^{13} - \frac{1}{250} a^{12} + \frac{3}{125} a^{11} + \frac{2}{125} a^{10} + \frac{27}{125} a^{9} - \frac{1}{50} a^{8} - \frac{34}{125} a^{7} + \frac{121}{250} a^{6} - \frac{53}{250} a^{5} + \frac{79}{250} a^{4} - \frac{1}{25} a^{3} + \frac{21}{125} a^{2} - \frac{37}{125} a + \frac{19}{50}$, $\frac{1}{1700188250} a^{15} + \frac{1568399}{1700188250} a^{14} + \frac{7666471}{850094125} a^{13} - \frac{12463017}{1700188250} a^{12} + \frac{31058557}{1700188250} a^{11} + \frac{36400811}{1700188250} a^{10} - \frac{364169193}{1700188250} a^{9} + \frac{106115151}{850094125} a^{8} + \frac{228812257}{1700188250} a^{7} + \frac{138247357}{340037650} a^{6} + \frac{46632419}{170018825} a^{5} - \frac{104527853}{1700188250} a^{4} - \frac{126105953}{1700188250} a^{3} - \frac{15658403}{1700188250} a^{2} + \frac{465025673}{1700188250} a - \frac{62504509}{170018825}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( -\frac{4408674}{11645125} a^{15} + \frac{9111867}{11645125} a^{14} - \frac{4828328}{2329025} a^{13} + \frac{64208763}{4658050} a^{12} - \frac{74865746}{2329025} a^{11} + \frac{604385423}{11645125} a^{10} - \frac{1355017396}{11645125} a^{9} + \frac{3042293409}{23290250} a^{8} + \frac{932340278}{11645125} a^{7} - \frac{2427145192}{11645125} a^{6} - \frac{1042613374}{11645125} a^{5} + \frac{10049437483}{23290250} a^{4} - \frac{5397942888}{11645125} a^{3} + \frac{3516821038}{11645125} a^{2} - \frac{1225678104}{11645125} a + \frac{56650863}{4658050} \) (order $4$)
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:  \( 5908.90081368 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_8$ (as 16T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2\times D_8$
Character table for $C_2\times D_8$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{-34}) \), 4.0.272.1, 4.0.4352.2, \(\Q(i, \sqrt{34})\), 8.0.321978368.4, 8.0.20123648.2, 8.0.5473632256.6

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$