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Properties

Label 16.4.81000012766...5625.1
Degree $16$
Signature $[4, 6]$
Discriminant $5^{8}\cdot 29^{10}\cdot 149^{4}$
Root discriminant $64.09$
Ramified primes $5, 29, 149$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 16T1439

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1017664, -1213216, -147792, 304792, -137384, 42256, -78873, 40198, -6768, 2338, -386, -818, 253, 20, -5, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 5*x^14 + 20*x^13 + 253*x^12 - 818*x^11 - 386*x^10 + 2338*x^9 - 6768*x^8 + 40198*x^7 - 78873*x^6 + 42256*x^5 - 137384*x^4 + 304792*x^3 - 147792*x^2 - 1213216*x + 1017664)
 
gp: K = bnfinit(x^16 - 5*x^15 - 5*x^14 + 20*x^13 + 253*x^12 - 818*x^11 - 386*x^10 + 2338*x^9 - 6768*x^8 + 40198*x^7 - 78873*x^6 + 42256*x^5 - 137384*x^4 + 304792*x^3 - 147792*x^2 - 1213216*x + 1017664, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 5 x^{14} + 20 x^{13} + 253 x^{12} - 818 x^{11} - 386 x^{10} + 2338 x^{9} - 6768 x^{8} + 40198 x^{7} - 78873 x^{6} + 42256 x^{5} - 137384 x^{4} + 304792 x^{3} - 147792 x^{2} - 1213216 x + 1017664 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(81000012766225321509609765625=5^{8}\cdot 29^{10}\cdot 149^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.09$
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:  $5, 29, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{26} a^{11} - \frac{9}{26} a^{10} + \frac{11}{26} a^{9} + \frac{4}{13} a^{8} + \frac{3}{26} a^{7} + \frac{1}{13} a^{6} - \frac{2}{13} a^{5} - \frac{1}{13} a^{4} + \frac{4}{13} a^{3} + \frac{1}{26} a + \frac{4}{13}$, $\frac{1}{52} a^{12} - \frac{1}{52} a^{11} - \frac{9}{52} a^{10} - \frac{2}{13} a^{9} - \frac{11}{52} a^{8} - \frac{1}{2} a^{7} - \frac{7}{26} a^{6} + \frac{9}{26} a^{5} - \frac{2}{13} a^{4} - \frac{7}{26} a^{3} - \frac{25}{52} a^{2} + \frac{4}{13} a + \frac{3}{13}$, $\frac{1}{1352} a^{13} - \frac{1}{1352} a^{12} - \frac{1}{104} a^{11} - \frac{45}{338} a^{10} - \frac{263}{1352} a^{9} - \frac{263}{676} a^{8} + \frac{9}{52} a^{7} - \frac{307}{676} a^{6} - \frac{64}{169} a^{5} - \frac{185}{676} a^{4} + \frac{151}{1352} a^{3} + \frac{121}{338} a^{2} - \frac{11}{338} a - \frac{17}{169}$, $\frac{1}{67600} a^{14} - \frac{23}{67600} a^{13} - \frac{407}{67600} a^{12} + \frac{157}{33800} a^{11} + \frac{9313}{67600} a^{10} - \frac{2819}{16900} a^{9} + \frac{9387}{33800} a^{8} - \frac{489}{33800} a^{7} - \frac{4187}{16900} a^{6} - \frac{113}{520} a^{5} + \frac{6463}{13520} a^{4} + \frac{2013}{33800} a^{3} + \frac{8377}{16900} a^{2} + \frac{33}{650} a - \frac{1581}{4225}$, $\frac{1}{2765285062689262595774446011486522400} a^{15} - \frac{3549327824028352068658410258469}{553057012537852519154889202297304480} a^{14} + \frac{271724625771078224368808336786599}{2765285062689262595774446011486522400} a^{13} + \frac{3096238221378888554414332022631321}{345660632836157824471805751435815300} a^{12} - \frac{1067426179576114135500863249923399}{553057012537852519154889202297304480} a^{11} + \frac{381937679069191351412385756304934769}{1382642531344631297887223005743261200} a^{10} + \frac{610653895494031180888787086515745423}{1382642531344631297887223005743261200} a^{9} - \frac{508710244363905486501733676832651703}{1382642531344631297887223005743261200} a^{8} - \frac{113281054727445625332293086840499229}{345660632836157824471805751435815300} a^{7} - \frac{232005336772253096832046912878787117}{1382642531344631297887223005743261200} a^{6} + \frac{90455390496578992277218360506481619}{553057012537852519154889202297304480} a^{5} + \frac{36752664985303345798794647078911149}{691321265672315648943611502871630600} a^{4} + \frac{7688609328306084799718731960860723}{86415158209039456117951437858953825} a^{3} + \frac{33244112289902066904662938940957108}{86415158209039456117951437858953825} a^{2} - \frac{902624006824095371933407999029820}{3456606328361578244718057514358153} a - \frac{14206779266163137996716392725986984}{86415158209039456117951437858953825}$

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:  $9$
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:  \( 91960024.8377 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1439:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2048
The 74 conjugacy class representatives for t16n1439 are not computed
Character table for t16n1439 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.65865543125.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{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}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$149$149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.1.2$x^{2} + 298$$2$$1$$1$$C_2$$[\ ]_{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
149.4.2.1$x^{4} + 745 x^{2} + 199809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$