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Properties

Label 16.0.14462138547...8496.6
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 23^{8}$
Root discriminant $66.45$
Ramified primes $2, 3, 23$
Class number $19200$ (GRH)
Class group $[4, 40, 120]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13286974, -8393144, 11026220, -5804824, 4326842, -1935192, 1046676, -396536, 169095, -54040, 18684, -4920, 1366, -280, 60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974)
 
gp: K = bnfinit(x^16 - 8*x^15 + 60*x^14 - 280*x^13 + 1366*x^12 - 4920*x^11 + 18684*x^10 - 54040*x^9 + 169095*x^8 - 396536*x^7 + 1046676*x^6 - 1935192*x^5 + 4326842*x^4 - 5804824*x^3 + 11026220*x^2 - 8393144*x + 13286974, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 60 x^{14} - 280 x^{13} + 1366 x^{12} - 4920 x^{11} + 18684 x^{10} - 54040 x^{9} + 169095 x^{8} - 396536 x^{7} + 1046676 x^{6} - 1935192 x^{5} + 4326842 x^{4} - 5804824 x^{3} + 11026220 x^{2} - 8393144 x + 13286974 \)

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:  \(144621385476274636765576298496=2^{48}\cdot 3^{8}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.45$
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 Galois and abelian over $\Q$.
Conductor:  \(1104=2^{4}\cdot 3\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{1104}(1,·)$, $\chi_{1104}(323,·)$, $\chi_{1104}(781,·)$, $\chi_{1104}(1103,·)$, $\chi_{1104}(275,·)$, $\chi_{1104}(277,·)$, $\chi_{1104}(599,·)$, $\chi_{1104}(1057,·)$, $\chi_{1104}(229,·)$, $\chi_{1104}(551,·)$, $\chi_{1104}(553,·)$, $\chi_{1104}(875,·)$, $\chi_{1104}(47,·)$, $\chi_{1104}(505,·)$, $\chi_{1104}(827,·)$, $\chi_{1104}(829,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{132864432006262313} a^{14} - \frac{1}{18980633143751759} a^{13} + \frac{41426465276959752}{132864432006262313} a^{12} + \frac{17170072350766205}{132864432006262313} a^{11} + \frac{59569301734124680}{132864432006262313} a^{10} - \frac{12357398531772736}{132864432006262313} a^{9} - \frac{33061884461306863}{132864432006262313} a^{8} - \frac{475795874927790}{18980633143751759} a^{7} + \frac{40278531490282985}{132864432006262313} a^{6} - \frac{17789385211336364}{132864432006262313} a^{5} - \frac{24699961133605624}{132864432006262313} a^{4} - \frac{4983221709105482}{132864432006262313} a^{3} + \frac{44398786038302834}{132864432006262313} a^{2} + \frac{26243697287447462}{132864432006262313} a + \frac{55870986316631942}{132864432006262313}$, $\frac{1}{114565412379335823417449} a^{15} + \frac{431129}{114565412379335823417449} a^{14} + \frac{45656246196777192557990}{114565412379335823417449} a^{13} - \frac{7110616324785738041498}{16366487482762260488207} a^{12} + \frac{18046996988071371627242}{114565412379335823417449} a^{11} - \frac{33554085979490955888802}{114565412379335823417449} a^{10} + \frac{7457240931626380320237}{16366487482762260488207} a^{9} - \frac{1211691337292201769189}{2437561965517783476967} a^{8} - \frac{6380970128639660430642}{114565412379335823417449} a^{7} - \frac{2831778595535907164318}{16366487482762260488207} a^{6} + \frac{39784324317352805738265}{114565412379335823417449} a^{5} + \frac{37172980449926097166057}{114565412379335823417449} a^{4} - \frac{48388811553834316791169}{114565412379335823417449} a^{3} + \frac{32549689688154966878655}{114565412379335823417449} a^{2} - \frac{1080397181588951833309}{114565412379335823417449} a - \frac{37924885709131430498882}{114565412379335823417449}$

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

Class group and class number

$C_{4}\times C_{40}\times C_{120}$, which has order $19200$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-138}) \), \(\Q(\sqrt{-46}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-69}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{3}, \sqrt{-46})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-23})\), \(\Q(\sqrt{2}, \sqrt{-69})\), \(\Q(\sqrt{6}, \sqrt{-23})\), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\sqrt{6}, \sqrt{-46})\), 4.0.9750528.5, 4.0.1083392.5, \(\Q(\zeta_{16})^+\), 4.4.18432.1, 8.0.1485512441856.5, 8.0.380291185115136.58, \(\Q(\zeta_{48})^+\), 8.0.380291185115136.75, 8.0.380291185115136.45, 8.0.95072796278784.65, 8.0.1173738225664.2

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 R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$23$23.4.2.1$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}$
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}$