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Properties

Label 16.0.17197312483...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{26}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}$
Root discriminant $283.28$
Ramified primes $2, 5, 7, 241$
Class number $240$ (GRH)
Class group $[4, 60]$ (GRH)
Galois group 16T1228

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5685401, 9719120, 1805439, -3908292, 1860427, 2788726, -1018960, -416602, 293810, -50506, 7952, -5034, 1443, -126, 11, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 11*x^14 - 126*x^13 + 1443*x^12 - 5034*x^11 + 7952*x^10 - 50506*x^9 + 293810*x^8 - 416602*x^7 - 1018960*x^6 + 2788726*x^5 + 1860427*x^4 - 3908292*x^3 + 1805439*x^2 + 9719120*x + 5685401)
 
gp: K = bnfinit(x^16 - 6*x^15 + 11*x^14 - 126*x^13 + 1443*x^12 - 5034*x^11 + 7952*x^10 - 50506*x^9 + 293810*x^8 - 416602*x^7 - 1018960*x^6 + 2788726*x^5 + 1860427*x^4 - 3908292*x^3 + 1805439*x^2 + 9719120*x + 5685401, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 11 x^{14} - 126 x^{13} + 1443 x^{12} - 5034 x^{11} + 7952 x^{10} - 50506 x^{9} + 293810 x^{8} - 416602 x^{7} - 1018960 x^{6} + 2788726 x^{5} + 1860427 x^{4} - 3908292 x^{3} + 1805439 x^{2} + 9719120 x + 5685401 \)

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:  \(1719731248304838654458056435302400000000=2^{26}\cdot 5^{8}\cdot 7^{8}\cdot 241^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $283.28$
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:  $2, 5, 7, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{198} a^{14} + \frac{2}{99} a^{13} + \frac{13}{198} a^{12} - \frac{49}{198} a^{11} - \frac{2}{11} a^{10} - \frac{67}{198} a^{9} - \frac{31}{99} a^{8} + \frac{29}{198} a^{7} + \frac{25}{99} a^{6} + \frac{9}{22} a^{5} + \frac{23}{99} a^{4} - \frac{49}{198} a^{3} - \frac{41}{198} a^{2} - \frac{1}{66} a + \frac{19}{198}$, $\frac{1}{56447482088847269217469216197995948466084960054} a^{15} + \frac{105167032234128629900092970067799723470134123}{56447482088847269217469216197995948466084960054} a^{14} - \frac{521172337812351101305439266173493043702803430}{28223741044423634608734608098997974233042480027} a^{13} + \frac{753644388783836834072785299748730676501652643}{9407913681474544869578202699665991411014160009} a^{12} - \frac{3527916169439066850701244598046083502457168079}{56447482088847269217469216197995948466084960054} a^{11} - \frac{3358031326142750666278051640304962737903255369}{56447482088847269217469216197995948466084960054} a^{10} + \frac{7199097365951716142709500661054550771161886931}{18815827362949089739156405399331982822028320018} a^{9} - \frac{4964100478057830579001266975266081969717704693}{18815827362949089739156405399331982822028320018} a^{8} + \frac{3818247643731281758808322319688823844284337399}{56447482088847269217469216197995948466084960054} a^{7} + \frac{2558573832068077861346890175567521135570563259}{5131589280804297201588110563454177133280450914} a^{6} + \frac{2755756786542377131102446691889745436579344031}{56447482088847269217469216197995948466084960054} a^{5} - \frac{2927973286289397536714797293742966989896780515}{18815827362949089739156405399331982822028320018} a^{4} - \frac{979142188029358350079835847360246765738724427}{3135971227158181623192734233221997137004720003} a^{3} - \frac{2810638052030493009345999075468829377165870676}{28223741044423634608734608098997974233042480027} a^{2} - \frac{26702888836367552221572276855245632904630886575}{56447482088847269217469216197995948466084960054} a - \frac{23003703178245160499641612482507444704035954025}{56447482088847269217469216197995948466084960054}$

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

Class group and class number

$C_{4}\times C_{60}$, which has order $240$ (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:  \( 127177365891 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1228:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 61 conjugacy class representatives for t16n1228 are not computed
Character table for t16n1228 is not computed

Intermediate fields

\(\Q(\sqrt{8435}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{1687}) \), 4.0.6025.1, 4.0.4723600.3, \(\Q(\sqrt{5}, \sqrt{1687})\), 8.0.1295926327833760000.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.18.1$x^{8} + 14 x^{6} + 10 x^{4} + 12 x^{2} + 16 x + 4$$4$$2$$18$$D_4\times C_2$$[2, 3, 7/2]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7Data not computed
241Data not computed