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Properties

Label 16.0.11939508427...0000.2
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 941^{4}$
Root discriminant $87.56$
Ramified primes $2, 5, 29, 941$
Class number $147680$ (GRH)
Class group $[4, 36920]$ (GRH)
Galois group 16T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2265405829, -247972336, 1092003662, -167301256, 273976211, -29815402, 38554411, -2727984, 3284104, -161942, 175264, -6642, 5830, -172, 113, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 113*x^14 - 172*x^13 + 5830*x^12 - 6642*x^11 + 175264*x^10 - 161942*x^9 + 3284104*x^8 - 2727984*x^7 + 38554411*x^6 - 29815402*x^5 + 273976211*x^4 - 167301256*x^3 + 1092003662*x^2 - 247972336*x + 2265405829)
 
gp: K = bnfinit(x^16 - 2*x^15 + 113*x^14 - 172*x^13 + 5830*x^12 - 6642*x^11 + 175264*x^10 - 161942*x^9 + 3284104*x^8 - 2727984*x^7 + 38554411*x^6 - 29815402*x^5 + 273976211*x^4 - 167301256*x^3 + 1092003662*x^2 - 247972336*x + 2265405829, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 113 x^{14} - 172 x^{13} + 5830 x^{12} - 6642 x^{11} + 175264 x^{10} - 161942 x^{9} + 3284104 x^{8} - 2727984 x^{7} + 38554411 x^{6} - 29815402 x^{5} + 273976211 x^{4} - 167301256 x^{3} + 1092003662 x^{2} - 247972336 x + 2265405829 \)

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:  \(11939508427262544380953600000000=2^{16}\cdot 5^{8}\cdot 29^{6}\cdot 941^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.56$
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:  $2, 5, 29, 941$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3906303824591788358806477930041317525338519146583405567501097} a^{15} + \frac{107736440876657145034083009884702985537638890394227349947323}{3906303824591788358806477930041317525338519146583405567501097} a^{14} + \frac{1278006674241222682699415534077867976465181355344744394147501}{3906303824591788358806477930041317525338519146583405567501097} a^{13} - \frac{1166247597152856967450149405981654216892703204912715151688087}{3906303824591788358806477930041317525338519146583405567501097} a^{12} + \frac{88336669406551166549711066102523405255437402156617443698027}{205594938136409913621393575265332501333606270872810819342163} a^{11} - \frac{802460102344511521619836615479248981310776694819709971717549}{3906303824591788358806477930041317525338519146583405567501097} a^{10} - \frac{1187208557522489256847956304870512534771466437405483489088524}{3906303824591788358806477930041317525338519146583405567501097} a^{9} - \frac{1374995907817744609755742448844225478440113035088101346465002}{3906303824591788358806477930041317525338519146583405567501097} a^{8} + \frac{1387400487210980515374433384313366303911319696481985078336575}{3906303824591788358806477930041317525338519146583405567501097} a^{7} + \frac{45731628797375433198298730143543180613330587376411979936483}{205594938136409913621393575265332501333606270872810819342163} a^{6} + \frac{389589635128548759688460842492159778980995408788028756372866}{3906303824591788358806477930041317525338519146583405567501097} a^{5} + \frac{228373880113297057775943467590457534992997167508072414639924}{3906303824591788358806477930041317525338519146583405567501097} a^{4} + \frac{26839117087963815261289668854076686033282040296840117112253}{3906303824591788358806477930041317525338519146583405567501097} a^{3} - \frac{1240572838957577532271111299793627827934057103337255220527899}{3906303824591788358806477930041317525338519146583405567501097} a^{2} - \frac{1731477165101393111952690778897341143872351546398818466607599}{3906303824591788358806477930041317525338519146583405567501097} a + \frac{1223469146584350414545814220966905793447519358649085550300800}{3906303824591788358806477930041317525338519146583405567501097}$

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

Class group and class number

$C_{4}\times C_{36920}$, which has order $147680$ (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:  \( 4024.00673312 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 44 conjugacy class representatives for t16n1025
Character table for t16n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.725.1, 8.0.3455359377440000.2, 8.0.3672007840000.1, 8.8.494613125.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
941Data not computed