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Properties

Label 16.0.48802261431...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 271181^{2}$
Root discriminant $71.70$
Ramified primes $5, 13, 29, 271181$
Class number $11520$ (GRH)
Class group $[6, 1920]$ (GRH)
Galois group 16T1177

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25104719, -12168844, 22963320, -16091218, 10063831, -4222836, 1901988, -586952, 222830, -52652, 18491, -3400, 1097, -144, 44, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 44*x^14 - 144*x^13 + 1097*x^12 - 3400*x^11 + 18491*x^10 - 52652*x^9 + 222830*x^8 - 586952*x^7 + 1901988*x^6 - 4222836*x^5 + 10063831*x^4 - 16091218*x^3 + 22963320*x^2 - 12168844*x + 25104719)
 
gp: K = bnfinit(x^16 - 4*x^15 + 44*x^14 - 144*x^13 + 1097*x^12 - 3400*x^11 + 18491*x^10 - 52652*x^9 + 222830*x^8 - 586952*x^7 + 1901988*x^6 - 4222836*x^5 + 10063831*x^4 - 16091218*x^3 + 22963320*x^2 - 12168844*x + 25104719, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 44 x^{14} - 144 x^{13} + 1097 x^{12} - 3400 x^{11} + 18491 x^{10} - 52652 x^{9} + 222830 x^{8} - 586952 x^{7} + 1901988 x^{6} - 4222836 x^{5} + 10063831 x^{4} - 16091218 x^{3} + 22963320 x^{2} - 12168844 x + 25104719 \)

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:  \(488022614316879794822906640625=5^{8}\cdot 13^{4}\cdot 29^{6}\cdot 271181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.70$
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:  $5, 13, 29, 271181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{22} a^{12} - \frac{3}{22} a^{11} + \frac{3}{22} a^{10} + \frac{3}{22} a^{9} + \frac{1}{22} a^{8} - \frac{1}{22} a^{7} + \frac{3}{11} a^{6} + \frac{1}{22} a^{5} - \frac{5}{22} a^{4} + \frac{3}{11} a^{3} + \frac{2}{11} a^{2} + \frac{5}{22} a + \frac{3}{11}$, $\frac{1}{22} a^{13} + \frac{5}{22} a^{11} + \frac{1}{22} a^{10} - \frac{1}{22} a^{9} + \frac{1}{11} a^{8} - \frac{4}{11} a^{7} - \frac{3}{22} a^{6} + \frac{9}{22} a^{5} + \frac{1}{11} a^{4} - \frac{1}{2} a^{3} + \frac{3}{11} a^{2} - \frac{1}{22} a - \frac{2}{11}$, $\frac{1}{108295330} a^{14} + \frac{253623}{54147665} a^{13} - \frac{453359}{108295330} a^{12} - \frac{10108361}{54147665} a^{11} - \frac{8346751}{108295330} a^{10} - \frac{22097469}{108295330} a^{9} - \frac{23113861}{108295330} a^{8} - \frac{1466739}{21659066} a^{7} + \frac{5357083}{108295330} a^{6} + \frac{31761683}{108295330} a^{5} + \frac{10588139}{108295330} a^{4} + \frac{10207305}{21659066} a^{3} + \frac{37219269}{108295330} a^{2} + \frac{10124586}{54147665} a - \frac{26070091}{54147665}$, $\frac{1}{20444273655614996661779264719238825670610} a^{15} + \frac{12086916863920857661404628515137}{4088854731122999332355852943847765134122} a^{14} - \frac{2408753097258940956516953042283038482}{157263643504730743552148190147990966697} a^{13} + \frac{45911819761636080743859214202909157886}{10222136827807498330889632359619412835305} a^{12} - \frac{1749031147311173530112754164714902271262}{10222136827807498330889632359619412835305} a^{11} + \frac{1454527679137784517591216417349854853676}{10222136827807498330889632359619412835305} a^{10} - \frac{2264368217173229714563685626870456185957}{20444273655614996661779264719238825670610} a^{9} - \frac{2352956229811875456800241169144379692227}{10222136827807498330889632359619412835305} a^{8} + \frac{5557960723494900381312945795880254137133}{20444273655614996661779264719238825670610} a^{7} + \frac{1158643692251509994766216680748532057359}{4088854731122999332355852943847765134122} a^{6} + \frac{7968934736472200468665763254115437827851}{20444273655614996661779264719238825670610} a^{5} - \frac{338806197073956139672094389489956768249}{20444273655614996661779264719238825670610} a^{4} + \frac{1682695603614087228626873919675402770162}{10222136827807498330889632359619412835305} a^{3} - \frac{6665620459144750201265705184828343034487}{20444273655614996661779264719238825670610} a^{2} + \frac{1040141703427927373391580659948472424408}{10222136827807498330889632359619412835305} a + \frac{2125448948723749200260969398176777340461}{10222136827807498330889632359619412835305}$

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

Class group and class number

$C_{6}\times C_{1920}$, which has order $11520$ (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:  \( 10968.6213178 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1177:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1024
The 76 conjugacy class representatives for t16n1177 are not computed
Character table for t16n1177 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.0.142539513125.1, 8.8.2576088125.1, 8.0.698586153825625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$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}$
5.4.2.1$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}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.3$x^{4} + 58$$4$$1$$3$$C_4$$[\ ]_{4}$
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}$
271181Data not computed