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Properties

Label 16.0.90014328985...2624.3
Degree $16$
Signature $[0, 8]$
Discriminant $2^{16}\cdot 13^{8}\cdot 17^{14}$
Root discriminant $86.03$
Ramified primes $2, 13, 17$
Class number $139264$ (GRH)
Class group $[4, 4, 4, 8, 272]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2317576561, -272656064, 993282715, -104591990, 198368818, -18462918, 24090615, -1943848, 1947249, -132122, 107531, -5828, 3980, -156, 91, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 91*x^14 - 156*x^13 + 3980*x^12 - 5828*x^11 + 107531*x^10 - 132122*x^9 + 1947249*x^8 - 1943848*x^7 + 24090615*x^6 - 18462918*x^5 + 198368818*x^4 - 104591990*x^3 + 993282715*x^2 - 272656064*x + 2317576561)
 
gp: K = bnfinit(x^16 - 2*x^15 + 91*x^14 - 156*x^13 + 3980*x^12 - 5828*x^11 + 107531*x^10 - 132122*x^9 + 1947249*x^8 - 1943848*x^7 + 24090615*x^6 - 18462918*x^5 + 198368818*x^4 - 104591990*x^3 + 993282715*x^2 - 272656064*x + 2317576561, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 91 x^{14} - 156 x^{13} + 3980 x^{12} - 5828 x^{11} + 107531 x^{10} - 132122 x^{9} + 1947249 x^{8} - 1943848 x^{7} + 24090615 x^{6} - 18462918 x^{5} + 198368818 x^{4} - 104591990 x^{3} + 993282715 x^{2} - 272656064 x + 2317576561 \)

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:  \(9001432898582155699240292122624=2^{16}\cdot 13^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.03$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(884=2^{2}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{884}(1,·)$, $\chi_{884}(259,·)$, $\chi_{884}(519,·)$, $\chi_{884}(781,·)$, $\chi_{884}(467,·)$, $\chi_{884}(727,·)$, $\chi_{884}(729,·)$, $\chi_{884}(155,·)$, $\chi_{884}(157,·)$, $\chi_{884}(417,·)$, $\chi_{884}(103,·)$, $\chi_{884}(365,·)$, $\chi_{884}(625,·)$, $\chi_{884}(883,·)$, $\chi_{884}(53,·)$, $\chi_{884}(831,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{9347967593006965880200401959963052993951829788617} a^{15} + \frac{4084114445089111147187698534120052899982138664401}{9347967593006965880200401959963052993951829788617} a^{14} - \frac{3908906287025226224286930599115958375782968573602}{9347967593006965880200401959963052993951829788617} a^{13} + \frac{3646302403394164440840256576172713496262964956600}{9347967593006965880200401959963052993951829788617} a^{12} - \frac{2291815155599577631697680258004251107209554273258}{9347967593006965880200401959963052993951829788617} a^{11} - \frac{1014013221765154162536516823613443469606539790174}{9347967593006965880200401959963052993951829788617} a^{10} - \frac{2459673929416682439348009582026488200568628004731}{9347967593006965880200401959963052993951829788617} a^{9} + \frac{602067814989698341434921759053809301576277404990}{9347967593006965880200401959963052993951829788617} a^{8} + \frac{4115605625624632691237893255613242342280962003304}{9347967593006965880200401959963052993951829788617} a^{7} + \frac{2042177877605285838934697532349515569590258446019}{9347967593006965880200401959963052993951829788617} a^{6} + \frac{4488536155889774446108469290821893518811223914288}{9347967593006965880200401959963052993951829788617} a^{5} - \frac{739886870622529324825880279128782263036010513849}{9347967593006965880200401959963052993951829788617} a^{4} - \frac{1664609384604064653010861215033007358634737920278}{9347967593006965880200401959963052993951829788617} a^{3} - \frac{121550900082912725331957220806456348097777647341}{9347967593006965880200401959963052993951829788617} a^{2} - \frac{2975425082420891628528546415272402271337712306613}{9347967593006965880200401959963052993951829788617} a - \frac{1194415634381393270188337321274838770290607785151}{9347967593006965880200401959963052993951829788617}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{8}\times C_{272}$, which has order $139264$ (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:  \( 3640.012213375973 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-13}, \sqrt{17})\), 4.4.4913.1, 4.0.13284752.4, 8.0.176484635701504.33, 8.0.3000238806925568.13, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$