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Properties

Label 16.0.40761133659...2256.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{40}\cdot 3^{12}\cdot 17^{8}$
Root discriminant $53.17$
Ramified primes $2, 3, 17$
Class number $1360$ (GRH)
Class group $[2, 2, 2, 170]$ (GRH)
Galois group $(C_2\times D_4):C_4$ (as 16T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![204241, 74100, 163352, 73728, 274372, 5796, 181556, -1920, 70945, 36, 14408, -216, 1534, -24, 68, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 68*x^14 - 24*x^13 + 1534*x^12 - 216*x^11 + 14408*x^10 + 36*x^9 + 70945*x^8 - 1920*x^7 + 181556*x^6 + 5796*x^5 + 274372*x^4 + 73728*x^3 + 163352*x^2 + 74100*x + 204241)
 
gp: K = bnfinit(x^16 + 68*x^14 - 24*x^13 + 1534*x^12 - 216*x^11 + 14408*x^10 + 36*x^9 + 70945*x^8 - 1920*x^7 + 181556*x^6 + 5796*x^5 + 274372*x^4 + 73728*x^3 + 163352*x^2 + 74100*x + 204241, 1)
 

Normalized defining polynomial

\( x^{16} + 68 x^{14} - 24 x^{13} + 1534 x^{12} - 216 x^{11} + 14408 x^{10} + 36 x^{9} + 70945 x^{8} - 1920 x^{7} + 181556 x^{6} + 5796 x^{5} + 274372 x^{4} + 73728 x^{3} + 163352 x^{2} + 74100 x + 204241 \)

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:  \(4076113365999630907663712256=2^{40}\cdot 3^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.17$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{51} a^{10} + \frac{2}{17} a^{9} - \frac{2}{17} a^{8} - \frac{4}{17} a^{7} + \frac{8}{51} a^{6} + \frac{6}{17} a^{5} + \frac{7}{51} a^{4} - \frac{4}{17} a^{3} + \frac{4}{17} a^{2} + \frac{3}{17} a - \frac{10}{51}$, $\frac{1}{51} a^{11} - \frac{8}{51} a^{9} + \frac{7}{51} a^{8} + \frac{4}{17} a^{7} + \frac{4}{51} a^{6} + \frac{1}{51} a^{5} - \frac{1}{17} a^{4} + \frac{16}{51} a^{3} + \frac{22}{51} a^{2} + \frac{7}{17} a - \frac{8}{51}$, $\frac{1}{153} a^{12} + \frac{7}{51} a^{9} + \frac{5}{51} a^{8} - \frac{8}{51} a^{7} + \frac{65}{153} a^{6} - \frac{7}{17} a^{5} - \frac{10}{51} a^{4} - \frac{2}{51} a^{3} - \frac{4}{17} a^{2} + \frac{10}{51} a - \frac{29}{153}$, $\frac{1}{153} a^{13} - \frac{1}{17} a^{9} - \frac{40}{153} a^{7} - \frac{3}{17} a^{6} + \frac{1}{3} a^{5} + \frac{4}{51} a^{3} - \frac{2}{17} a^{2} + \frac{37}{153} a - \frac{5}{17}$, $\frac{1}{1071} a^{14} + \frac{1}{357} a^{13} - \frac{2}{1071} a^{12} + \frac{2}{357} a^{11} + \frac{1}{119} a^{10} + \frac{31}{357} a^{9} - \frac{8}{63} a^{8} + \frac{157}{357} a^{7} + \frac{467}{1071} a^{6} + \frac{101}{357} a^{5} - \frac{2}{17} a^{4} + \frac{106}{357} a^{3} - \frac{362}{1071} a^{2} + \frac{44}{119} a + \frac{1}{1071}$, $\frac{1}{269620600702790069976261737129859} a^{15} - \frac{13483707952303610717770184615}{89873533567596689992087245709953} a^{14} + \frac{154904100191490025667826357611}{89873533567596689992087245709953} a^{13} - \frac{868779056695984336602419912468}{269620600702790069976261737129859} a^{12} - \frac{716602758250764201960492622457}{89873533567596689992087245709953} a^{11} - \frac{243309364875303237712771438072}{89873533567596689992087245709953} a^{10} - \frac{8706524507337123433223233224445}{269620600702790069976261737129859} a^{9} + \frac{1484943858476781290417134956213}{29957844522532229997362415236651} a^{8} - \frac{700441810892528725383259091531}{4279692074647461428194630748093} a^{7} + \frac{10427458363148034181861237952864}{38517228671827152853751676732837} a^{6} + \frac{24943557047908967918782266179150}{89873533567596689992087245709953} a^{5} - \frac{39294489739459996192186743614570}{89873533567596689992087245709953} a^{4} - \frac{112750297226578947242949874477394}{269620600702790069976261737129859} a^{3} + \frac{4525758375803892337246512612715}{29957844522532229997362415236651} a^{2} + \frac{36830063350803986326915572593030}{89873533567596689992087245709953} a + \frac{89911392633446366238014183803477}{269620600702790069976261737129859}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{170}$, which has order $1360$ (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:  \( 9072.35800888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_4$ (as 16T120):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times D_4):C_4$
Character table for $(C_2\times D_4):C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.9792.1, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.1534132224.1

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.6.2$x^{8} + 85 x^{4} + 2601$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$