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Properties

Label 16.8.18346075763...5504.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{34}\cdot 17^{8}\cdot 11525081^{3}$
Root discriminant $379.29$
Ramified primes $2, 17, 11525081$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 16T1472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![168456667648, 11943711488, -56144174560, -3786981696, 8825662592, 601499728, -833622374, -60179612, 46886345, 3401900, -1539484, -102876, 28364, 1516, -266, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 - 266*x^14 + 1516*x^13 + 28364*x^12 - 102876*x^11 - 1539484*x^10 + 3401900*x^9 + 46886345*x^8 - 60179612*x^7 - 833622374*x^6 + 601499728*x^5 + 8825662592*x^4 - 3786981696*x^3 - 56144174560*x^2 + 11943711488*x + 168456667648)
 
gp: K = bnfinit(x^16 - 8*x^15 - 266*x^14 + 1516*x^13 + 28364*x^12 - 102876*x^11 - 1539484*x^10 + 3401900*x^9 + 46886345*x^8 - 60179612*x^7 - 833622374*x^6 + 601499728*x^5 + 8825662592*x^4 - 3786981696*x^3 - 56144174560*x^2 + 11943711488*x + 168456667648, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} - 266 x^{14} + 1516 x^{13} + 28364 x^{12} - 102876 x^{11} - 1539484 x^{10} + 3401900 x^{9} + 46886345 x^{8} - 60179612 x^{7} - 833622374 x^{6} + 601499728 x^{5} + 8825662592 x^{4} - 3786981696 x^{3} - 56144174560 x^{2} + 11943711488 x + 168456667648 \)

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:  \(183460757637086369287254273534720689045504=2^{34}\cdot 17^{8}\cdot 11525081^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $379.29$
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:  $2, 17, 11525081$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{64} a^{13} - \frac{3}{32} a^{11} + \frac{3}{16} a^{10} + \frac{5}{16} a^{9} + \frac{5}{16} a^{8} + \frac{5}{16} a^{7} + \frac{7}{16} a^{6} - \frac{7}{64} a^{5} + \frac{7}{16} a^{4} + \frac{15}{32} a^{3} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{640} a^{14} - \frac{1}{320} a^{13} + \frac{13}{320} a^{12} + \frac{3}{80} a^{11} - \frac{17}{160} a^{10} - \frac{37}{160} a^{9} + \frac{59}{160} a^{8} - \frac{19}{160} a^{7} - \frac{127}{640} a^{6} - \frac{139}{320} a^{5} - \frac{25}{64} a^{4} + \frac{49}{160} a^{3} + \frac{31}{80} a^{2} - \frac{9}{40} a - \frac{3}{10}$, $\frac{1}{14079343498870353106538384766599896602244920487721287856677931520} a^{15} + \frac{533096155331942138272317812425067298357353288294885743555}{7332991405661642242988742065937446147002562754021504092019756} a^{14} - \frac{50511038419244333905217524051095067391000913578751154412376949}{7039671749435176553269192383299948301122460243860643928338965760} a^{13} + \frac{12327478262749007325624707825381519174598466455387672307241513}{1173278624905862758878198730549991383520410040643440654723160960} a^{12} + \frac{36187969830481811524411593991821071142895696205520040888560799}{703967174943517655326919238329994830112246024386064392833896576} a^{11} + \frac{13200479697384483486387772222861822989410413311668818872223213}{270756605747506790510353553203844165427786932456178612628421760} a^{10} - \frac{31233424850199450996701820423640065237842892054113118637954213}{100566739277645379332417034047142118587463717769437770404842368} a^{9} + \frac{1673949002090453181453620619776863589975825969331012432951161059}{3519835874717588276634596191649974150561230121930321964169482880} a^{8} - \frac{6339155906691995321240827800867098617940225573430003111757368599}{14079343498870353106538384766599896602244920487721287856677931520} a^{7} - \frac{180694915874376748026628120150776436372226533603491636212875031}{1173278624905862758878198730549991383520410040643440654723160960} a^{6} - \frac{271004094872711632231939892746028001045714807159301429184607763}{7039671749435176553269192383299948301122460243860643928338965760} a^{5} + \frac{19379831479703786025587010212509002429142147587297355833837439}{97773218742155229906516560879165948626700836720286721226930080} a^{4} + \frac{292899918833583312021695864140061461993928571803172958676423}{1692228785921917440689709707524026033923668327851116328927636} a^{3} - \frac{181362325363687463443769520165240918454565507383128401276147}{5714019277138942007523695116314893101560438509627146045729680} a^{2} + \frac{39635475623267199271268370983388735687077510845133410984300333}{87995896867939706915864904791249353764030753048258049104237072} a - \frac{35149937147338263178322851483295681620213538510215287361875613}{109994871084924633644831130989061692205038441310322561380296340}$

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

Class group and class number

$C_{2}\times C_{2}$, 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:  $11$
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:  \( 4988064436360000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1472:

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 t16n1472 are not computed
Character table for t16n1472 is not computed

Intermediate fields

\(\Q(\sqrt{34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{2}, \sqrt{17})\), 8.8.15771013778653184.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.22.84$x^{8} + 4 x^{7} + 10 x^{4} + 4 x^{2} + 14$$8$$1$$22$$D_4$$[2, 3, 7/2]$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11525081Data not computed