↑

Properties

Label 16.0.45960213837...7121.1
Degree $16$
Signature $[0, 8]$
Discriminant $41^{14}\cdot 59^{4}$
Root discriminant $71.43$
Ramified primes $41, 59$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2 : C_8$ (as 16T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12113, -11559, 81805, 129812, 371280, 473927, 107105, -27890, 24141, 8153, 3511, 108, -146, -84, -2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113)
 
gp: K = bnfinit(x^16 - 2*x^15 - 2*x^14 - 84*x^13 - 146*x^12 + 108*x^11 + 3511*x^10 + 8153*x^9 + 24141*x^8 - 27890*x^7 + 107105*x^6 + 473927*x^5 + 371280*x^4 + 129812*x^3 + 81805*x^2 - 11559*x + 12113, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 2 x^{14} - 84 x^{13} - 146 x^{12} + 108 x^{11} + 3511 x^{10} + 8153 x^{9} + 24141 x^{8} - 27890 x^{7} + 107105 x^{6} + 473927 x^{5} + 371280 x^{4} + 129812 x^{3} + 81805 x^{2} - 11559 x + 12113 \)

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:  \(459602138371809190522512417121=41^{14}\cdot 59^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.43$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a$, $\frac{1}{1328} a^{14} - \frac{1}{1328} a^{13} - \frac{31}{664} a^{12} + \frac{245}{1328} a^{11} - \frac{227}{1328} a^{10} + \frac{183}{664} a^{9} - \frac{329}{664} a^{8} - \frac{155}{1328} a^{7} - \frac{17}{83} a^{6} + \frac{239}{1328} a^{5} + \frac{55}{332} a^{4} + \frac{113}{664} a^{3} + \frac{149}{664} a^{2} + \frac{36}{83} a - \frac{605}{1328}$, $\frac{1}{55055489327886102879853150694462029839536} a^{15} + \frac{1814371293770416147789797247481816377}{55055489327886102879853150694462029839536} a^{14} - \frac{940827392199830959068487450718611687331}{27527744663943051439926575347231014919768} a^{13} + \frac{1257961053229916702899538611061806424887}{55055489327886102879853150694462029839536} a^{12} - \frac{1401665215533209535480686910655800952527}{55055489327886102879853150694462029839536} a^{11} - \frac{2276622040825253194445787217933144950819}{27527744663943051439926575347231014919768} a^{10} + \frac{9563029238734193405365113612333783625577}{27527744663943051439926575347231014919768} a^{9} - \frac{11700721216915262849275288751953751957583}{55055489327886102879853150694462029839536} a^{8} - \frac{7523037866950661323624376695441840778075}{27527744663943051439926575347231014919768} a^{7} - \frac{24741040649118912832551655448081824302091}{55055489327886102879853150694462029839536} a^{6} + \frac{10574065495798471107134778614873885239883}{27527744663943051439926575347231014919768} a^{5} - \frac{6504965962995413522013208835019315958349}{27527744663943051439926575347231014919768} a^{4} - \frac{660422020568494174593647474032579456949}{3440968082992881429990821918403876864971} a^{3} - \frac{46530283803462865070866283067313174703}{331659574264374113734055124665433914696} a^{2} - \frac{11686068355226042918487825553935009467965}{55055489327886102879853150694462029839536} a + \frac{1950302170297555082590579621853752802951}{6881936165985762859981643836807753729942}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 67708958.4741 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_8$ (as 16T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 : C_8$
Character table for $C_2^2 : C_8$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.2.99179.1, 4.4.68921.1, 4.2.4066339.1, 8.0.194754273881.1, 8.4.677939627379761.1, 8.4.16535112862921.1

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

Sibling fields

Galois closure: data not computed
Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ R

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
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{59}$$x + 3$$1$$1$$0$Trivial$[\ ]$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$