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Properties

Label 16.8.30417192973...5625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{12}\cdot 11^{3}\cdot 101^{6}\cdot 4451^{3}$
Root discriminant $142.95$
Ramified primes $5, 11, 101, 4451$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1643

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2658144721, -564236981, -927639173, 105176879, 27981317, 3385851, 13887186, -1596582, -581363, 116979, -51646, -1108, 207, -123, 38, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 38*x^14 - 123*x^13 + 207*x^12 - 1108*x^11 - 51646*x^10 + 116979*x^9 - 581363*x^8 - 1596582*x^7 + 13887186*x^6 + 3385851*x^5 + 27981317*x^4 + 105176879*x^3 - 927639173*x^2 - 564236981*x + 2658144721)
 
gp: K = bnfinit(x^16 - 2*x^15 + 38*x^14 - 123*x^13 + 207*x^12 - 1108*x^11 - 51646*x^10 + 116979*x^9 - 581363*x^8 - 1596582*x^7 + 13887186*x^6 + 3385851*x^5 + 27981317*x^4 + 105176879*x^3 - 927639173*x^2 - 564236981*x + 2658144721, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 38 x^{14} - 123 x^{13} + 207 x^{12} - 1108 x^{11} - 51646 x^{10} + 116979 x^{9} - 581363 x^{8} - 1596582 x^{7} + 13887186 x^{6} + 3385851 x^{5} + 27981317 x^{4} + 105176879 x^{3} - 927639173 x^{2} - 564236981 x + 2658144721 \)

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:  \(30417192973888736937325752197265625=5^{12}\cdot 11^{3}\cdot 101^{6}\cdot 4451^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.95$
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:  $5, 11, 101, 4451$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1595} a^{14} + \frac{96}{1595} a^{13} + \frac{60}{319} a^{12} - \frac{199}{1595} a^{11} - \frac{111}{319} a^{10} + \frac{486}{1595} a^{9} - \frac{8}{145} a^{8} + \frac{204}{1595} a^{7} - \frac{558}{1595} a^{6} - \frac{15}{319} a^{5} - \frac{386}{1595} a^{4} + \frac{218}{1595} a^{3} - \frac{113}{1595} a^{2} - \frac{598}{1595} a + \frac{416}{1595}$, $\frac{1}{2016889728832925107303324900520324403518227854113607084258144775} a^{15} - \frac{31922538783427567606706593745377628848816328069663078154313}{2016889728832925107303324900520324403518227854113607084258144775} a^{14} + \frac{450341900201647987836143631179703557809383874677559917627816531}{2016889728832925107303324900520324403518227854113607084258144775} a^{13} + \frac{449866124201970235479191895828332493775862993790299054632573686}{2016889728832925107303324900520324403518227854113607084258144775} a^{12} - \frac{959026026606553934030208780223433052311120114589184209511543139}{2016889728832925107303324900520324403518227854113607084258144775} a^{11} - \frac{496761272657773907045038877302292289051229927296568973322259379}{2016889728832925107303324900520324403518227854113607084258144775} a^{10} + \frac{237787145984933560213804426820912688807258227348189829932948998}{2016889728832925107303324900520324403518227854113607084258144775} a^{9} - \frac{603388182259766227276129006366734647880209403373784865600910024}{2016889728832925107303324900520324403518227854113607084258144775} a^{8} - \frac{977136284314109785954678618833325936900193818216211306703051874}{2016889728832925107303324900520324403518227854113607084258144775} a^{7} - \frac{881619792857693139730639626089575687080331993631774813897769718}{2016889728832925107303324900520324403518227854113607084258144775} a^{6} + \frac{520267176018923023843401858453622955047700084701588825414490059}{2016889728832925107303324900520324403518227854113607084258144775} a^{5} + \frac{50500272983169828015485222719280628397376741495763305956120002}{2016889728832925107303324900520324403518227854113607084258144775} a^{4} + \frac{151504723134897948151537027287564731666707399412679737045022689}{403377945766585021460664980104064880703645570822721416851628955} a^{3} + \frac{538163153157897677017042307189426558971059875858549532718114609}{2016889728832925107303324900520324403518227854113607084258144775} a^{2} + \frac{419027354581876850002772651371128487063620797403844452883452228}{2016889728832925107303324900520324403518227854113607084258144775} a - \frac{717183701012324946977980381309922970019686801420186205255644814}{2016889728832925107303324900520324403518227854113607084258144775}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 243134769063 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1

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 $16$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
101.8.4.1$x^{8} + 244824 x^{4} - 1030301 x^{2} + 14984697744$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
4451Data not computed