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Properties

Label 16.8.57573646968...1776.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{4}\cdot 107^{4}\cdot 139^{4}$
Root discriminant $1983.87$
Ramified primes $2, 3, 13, 29, 107, 139$
Class number $64$ (GRH)
Class group $[2, 2, 2, 2, 2, 2]$ (GRH)
Galois group $C_2^2.C_2^5.C_2$ (as 16T493)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52639955977261288880155041, 0, -2145283046375480686894752, 0, 3478561065005540201730, 0, 3271348226715498720, 0, -3729301891827816, 0, -1020008167440, 0, 11838306, 0, 17472, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 17472*x^14 + 11838306*x^12 - 1020008167440*x^10 - 3729301891827816*x^8 + 3271348226715498720*x^6 + 3478561065005540201730*x^4 - 2145283046375480686894752*x^2 + 52639955977261288880155041)
 
gp: K = bnfinit(x^16 + 17472*x^14 + 11838306*x^12 - 1020008167440*x^10 - 3729301891827816*x^8 + 3271348226715498720*x^6 + 3478561065005540201730*x^4 - 2145283046375480686894752*x^2 + 52639955977261288880155041, 1)
 

Normalized defining polynomial

\( x^{16} + 17472 x^{14} + 11838306 x^{12} - 1020008167440 x^{10} - 3729301891827816 x^{8} + 3271348226715498720 x^{6} + 3478561065005540201730 x^{4} - 2145283046375480686894752 x^{2} + 52639955977261288880155041 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(57573646968444953741950777740415027177610419976011776=2^{56}\cdot 3^{14}\cdot 13^{6}\cdot 29^{4}\cdot 107^{4}\cdot 139^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1983.87$
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:  $2, 3, 13, 29, 107, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{417} a^{8} - \frac{41}{139} a^{6} + \frac{16}{139} a^{4} + \frac{3}{139} a^{2}$, $\frac{1}{417} a^{9} - \frac{41}{139} a^{7} + \frac{16}{139} a^{5} + \frac{3}{139} a^{3}$, $\frac{1}{3881853} a^{10} - \frac{382}{1293951} a^{8} - \frac{35047}{1293951} a^{6} - \frac{142438}{1293951} a^{4} + \frac{593380}{1293951} a^{2}$, $\frac{1}{3881853} a^{11} - \frac{382}{1293951} a^{9} - \frac{35047}{1293951} a^{7} - \frac{142438}{1293951} a^{5} + \frac{593380}{1293951} a^{3}$, $\frac{1}{1674309190401} a^{12} + \frac{5824}{558103063467} a^{10} + \frac{3946102}{558103063467} a^{8} - \frac{153968367991}{558103063467} a^{6} + \frac{166960440715}{558103063467} a^{4} - \frac{202210}{431317} a^{2}$, $\frac{1}{1674309190401} a^{13} + \frac{5824}{558103063467} a^{11} + \frac{3946102}{558103063467} a^{9} - \frac{153968367991}{558103063467} a^{7} + \frac{166960440715}{558103063467} a^{5} - \frac{202210}{431317} a^{3}$, $\frac{1}{10858495995021692916028620305773018370911707817469839432119214564945197116889} a^{14} + \frac{16900725184079865117938732891092835082422148109896820554998413}{75933538426725125286913428711699429167214739982306569455379122831784595223} a^{12} + \frac{1073143887691115205218931277857800890986684321901263580327147459395892}{10858495995021692916028620305773018370911707817469839432119214564945197116889} a^{10} - \frac{231077775242022538013204399014697285571930142853540378566091421300860135}{1206499555002410324003180033974779818990189757496648825791023840549466346321} a^{8} - \frac{45563768931165728753878316954594619662883517548772256913756160251489878537}{3619498665007230972009540101924339456970569272489946477373071521648399038963} a^{6} + \frac{59501105830217732474832624884169777978835099992935849881962727396738}{226803692512557295258885598090194993408971934575112245929677201840347} a^{4} + \frac{7025011019908691235805285001844672034343133968132055652870476454}{19456076674382460984794865793226825643626292446806300469024073867} a^{2} - \frac{221971525317174309877583660085200748075409898620318162918}{1156629024317616888999712198900102544819126276913844643209}$, $\frac{1}{10858495995021692916028620305773018370911707817469839432119214564945197116889} a^{15} + \frac{16900725184079865117938732891092835082422148109896820554998413}{75933538426725125286913428711699429167214739982306569455379122831784595223} a^{13} + \frac{1073143887691115205218931277857800890986684321901263580327147459395892}{10858495995021692916028620305773018370911707817469839432119214564945197116889} a^{11} - \frac{231077775242022538013204399014697285571930142853540378566091421300860135}{1206499555002410324003180033974779818990189757496648825791023840549466346321} a^{9} - \frac{45563768931165728753878316954594619662883517548772256913756160251489878537}{3619498665007230972009540101924339456970569272489946477373071521648399038963} a^{7} + \frac{59501105830217732474832624884169777978835099992935849881962727396738}{226803692512557295258885598090194993408971934575112245929677201840347} a^{5} + \frac{7025011019908691235805285001844672034343133968132055652870476454}{19456076674382460984794865793226825643626292446806300469024073867} a^{3} - \frac{221971525317174309877583660085200748075409898620318162918}{1156629024317616888999712198900102544819126276913844643209} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $64$ (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:  $11$
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:  \( 11036207007800000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.C_2^5.C_2$ (as 16T493):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$
Character table for $C_2^2.C_2^5.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.7488.1, 4.4.179712.2, 4.4.13824.1, 8.8.129185611776.1

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
3Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
29.4.2.2$x^{4} - 29 x^{2} + 2523$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
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}$
$107$$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{107}$$x + 3$$1$$1$$0$Trivial$[\ ]$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.2.1.2$x^{2} + 321$$2$$1$$1$$C_2$$[\ ]_{2}$
107.4.2.2$x^{4} - 107 x^{2} + 57245$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
107.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$139$139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$