↑

Properties

Label 16.0.22144295318...8144.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{12}\cdot 23^{6}$
Root discriminant $59.10$
Ramified primes $2, 3, 23$
Class number $256$ (GRH)
Class group $[2, 4, 4, 8]$ (GRH)
Galois group $C_4.D_4$ (as 16T30)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![855742, 1020648, 989808, -963552, 576784, -409752, 166488, -73872, 32370, -7104, 4236, -552, 388, -24, 24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 24*x^14 - 24*x^13 + 388*x^12 - 552*x^11 + 4236*x^10 - 7104*x^9 + 32370*x^8 - 73872*x^7 + 166488*x^6 - 409752*x^5 + 576784*x^4 - 963552*x^3 + 989808*x^2 + 1020648*x + 855742)
 
gp: K = bnfinit(x^16 + 24*x^14 - 24*x^13 + 388*x^12 - 552*x^11 + 4236*x^10 - 7104*x^9 + 32370*x^8 - 73872*x^7 + 166488*x^6 - 409752*x^5 + 576784*x^4 - 963552*x^3 + 989808*x^2 + 1020648*x + 855742, 1)
 

Normalized defining polynomial

\( x^{16} + 24 x^{14} - 24 x^{13} + 388 x^{12} - 552 x^{11} + 4236 x^{10} - 7104 x^{9} + 32370 x^{8} - 73872 x^{7} + 166488 x^{6} - 409752 x^{5} + 576784 x^{4} - 963552 x^{3} + 989808 x^{2} + 1020648 x + 855742 \)

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:  \(22144295318673432094540038144=2^{48}\cdot 3^{12}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.10$
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:  $2, 3, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{143} a^{12} - \frac{3}{11} a^{11} + \frac{34}{143} a^{10} - \frac{10}{143} a^{9} - \frac{1}{13} a^{8} - \frac{7}{143} a^{7} + \frac{1}{13} a^{6} - \frac{42}{143} a^{5} + \frac{68}{143} a^{4} + \frac{2}{11} a^{3} + \frac{7}{143} a^{2} - \frac{47}{143} a - \frac{28}{143}$, $\frac{1}{143} a^{13} - \frac{57}{143} a^{11} + \frac{29}{143} a^{10} + \frac{28}{143} a^{9} - \frac{7}{143} a^{8} + \frac{24}{143} a^{7} - \frac{42}{143} a^{6} + \frac{3}{143} a^{5} - \frac{3}{11} a^{4} + \frac{20}{143} a^{3} - \frac{60}{143} a^{2} - \frac{2}{143} a + \frac{4}{11}$, $\frac{1}{2746028857} a^{14} + \frac{8619222}{2746028857} a^{13} + \frac{2228507}{2746028857} a^{12} - \frac{19503288}{249638987} a^{11} - \frac{80807488}{2746028857} a^{10} - \frac{646776538}{2746028857} a^{9} + \frac{769440273}{2746028857} a^{8} - \frac{446402260}{2746028857} a^{7} + \frac{1558230}{9184043} a^{6} + \frac{1196009571}{2746028857} a^{5} - \frac{131059935}{2746028857} a^{4} + \frac{1365665534}{2746028857} a^{3} + \frac{953625065}{2746028857} a^{2} - \frac{1364536214}{2746028857} a + \frac{84848416}{211232989}$, $\frac{1}{1793808919447721251190339171887511} a^{15} + \frac{23464827539211642965698}{1793808919447721251190339171887511} a^{14} - \frac{1540604018156110910819465888952}{1793808919447721251190339171887511} a^{13} - \frac{845945421997808094871077116693}{1793808919447721251190339171887511} a^{12} + \frac{553840809409164610875202862533249}{1793808919447721251190339171887511} a^{11} + \frac{25223435196134858270894803898711}{163073538131611022835485379262501} a^{10} + \frac{512880275966006818022397369315023}{1793808919447721251190339171887511} a^{9} - \frac{15614328405016051815731244241203}{77991692149900923964797355299457} a^{8} + \frac{688717098934763187802614578141988}{1793808919447721251190339171887511} a^{7} - \frac{516039157803343208326730836112521}{1793808919447721251190339171887511} a^{6} + \frac{785373398048980628376515283880704}{1793808919447721251190339171887511} a^{5} + \frac{62122277673212784087712928765319}{137985301495978557783872243991347} a^{4} - \frac{225906657349560657419920768392367}{1793808919447721251190339171887511} a^{3} + \frac{527827278039969799192808717327365}{1793808919447721251190339171887511} a^{2} - \frac{742272143936610848245146866677072}{1793808919447721251190339171887511} a + \frac{87103834109594605763120968995862}{1793808919447721251190339171887511}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{8}$, which has order $256$ (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:  \( 204141.742403 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4$ (as 16T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_4.D_4$
Character table for $C_4.D_4$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{6}) \), 4.4.105984.2, 4.4.105984.1, \(\Q(\sqrt{2}, \sqrt{3})\), 8.0.404373897216.34, 8.8.179721732096.1, 8.0.1617495588864.18

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$3$3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$