↑

Properties

Label 16.8.21767823360...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 3^{12}\cdot 5^{12}$
Root discriminant $21.56$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_4 \times D_4$ (as 16T19)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 33, -66, 116, 24, -55, -316, 123, 460, -253, -180, 122, 18, -15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 15*x^14 + 18*x^13 + 122*x^12 - 180*x^11 - 253*x^10 + 460*x^9 + 123*x^8 - 316*x^7 - 55*x^6 + 24*x^5 + 116*x^4 - 66*x^3 + 33*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^16 - 2*x^15 - 15*x^14 + 18*x^13 + 122*x^12 - 180*x^11 - 253*x^10 + 460*x^9 + 123*x^8 - 316*x^7 - 55*x^6 + 24*x^5 + 116*x^4 - 66*x^3 + 33*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 15 x^{14} + 18 x^{13} + 122 x^{12} - 180 x^{11} - 253 x^{10} + 460 x^{9} + 123 x^{8} - 316 x^{7} - 55 x^{6} + 24 x^{5} + 116 x^{4} - 66 x^{3} + 33 x^{2} - 10 x + 1 \)

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:  \(2176782336000000000000=2^{24}\cdot 3^{12}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.56$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{211} a^{14} + \frac{41}{211} a^{13} + \frac{50}{211} a^{12} + \frac{70}{211} a^{11} + \frac{100}{211} a^{10} + \frac{44}{211} a^{9} + \frac{6}{211} a^{8} + \frac{67}{211} a^{7} - \frac{10}{211} a^{6} + \frac{61}{211} a^{5} - \frac{75}{211} a^{4} - \frac{13}{211} a^{3} + \frac{96}{211} a^{2} - \frac{28}{211} a - \frac{21}{211}$, $\frac{1}{1242923141} a^{15} - \frac{792625}{1242923141} a^{14} - \frac{460131901}{1242923141} a^{13} - \frac{56903791}{1242923141} a^{12} + \frac{423812429}{1242923141} a^{11} - \frac{367548260}{1242923141} a^{10} + \frac{422828122}{1242923141} a^{9} + \frac{115228588}{1242923141} a^{8} - \frac{356749870}{1242923141} a^{7} - \frac{449763085}{1242923141} a^{6} + \frac{260391994}{1242923141} a^{5} - \frac{176938849}{1242923141} a^{4} + \frac{20867625}{1242923141} a^{3} + \frac{467239164}{1242923141} a^{2} + \frac{505724548}{1242923141} a + \frac{460976116}{1242923141}$

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:  \( 67733.2873693 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times D_4$ (as 16T19):

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 $C_4 \times D_4$
Character table for $C_4 \times D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{15})^+\), 4.4.8000.1, 4.2.8640.1, \(\Q(\sqrt{5}, \sqrt{6})\), 4.2.8640.2, 8.4.1866240000.2, 8.8.5184000000.2, 8.4.46656000000.2

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 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
3Data not computed
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$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}$