↑

Properties

Label 16.0.596...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.964\times 10^{23}$
Root discriminant \(30.62\)
Ramified primes $2,5,13$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81)
 
gp: K = bnfinit(y^16 - 9*y^14 + 135*y^12 - 90*y^10 + 663*y^8 - 270*y^6 + 1150*y^4 + 108*y^2 + 81, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81)
 

\( x^{16} - 9x^{14} + 135x^{12} - 90x^{10} + 663x^{8} - 270x^{6} + 1150x^{4} + 108x^{2} + 81 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(596430979135513600000000\) \(\medspace = 2^{16}\cdot 5^{8}\cdot 13^{12}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(30.62\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 5^{1/2}13^{3/4}\approx 30.61769638004527$
Ramified primes:   \(2\), \(5\), \(13\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(260=2^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(131,·)$, $\chi_{260}(129,·)$, $\chi_{260}(79,·)$, $\chi_{260}(209,·)$, $\chi_{260}(259,·)$, $\chi_{260}(21,·)$, $\chi_{260}(151,·)$, $\chi_{260}(31,·)$, $\chi_{260}(161,·)$, $\chi_{260}(99,·)$, $\chi_{260}(229,·)$, $\chi_{260}(109,·)$, $\chi_{260}(239,·)$, $\chi_{260}(51,·)$, $\chi_{260}(181,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{6}-\frac{2}{9}a^{2}$, $\frac{1}{27}a^{11}-\frac{1}{9}a^{9}+\frac{2}{27}a^{7}-\frac{1}{9}a^{5}+\frac{10}{27}a^{3}$, $\frac{1}{27}a^{12}+\frac{2}{27}a^{8}+\frac{1}{9}a^{6}+\frac{10}{27}a^{4}+\frac{1}{9}a^{2}$, $\frac{1}{27}a^{13}+\frac{2}{27}a^{9}+\frac{1}{9}a^{7}+\frac{1}{27}a^{5}+\frac{1}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{3922918479}a^{14}-\frac{2456099}{3922918479}a^{12}-\frac{31374718}{3922918479}a^{10}+\frac{371689871}{3922918479}a^{8}+\frac{618090241}{3922918479}a^{6}+\frac{502427488}{3922918479}a^{4}+\frac{67067113}{1307639493}a^{2}+\frac{37341311}{145293277}$, $\frac{1}{3922918479}a^{15}-\frac{2456099}{3922918479}a^{13}-\frac{31374718}{3922918479}a^{11}+\frac{371689871}{3922918479}a^{9}+\frac{618090241}{3922918479}a^{7}+\frac{502427488}{3922918479}a^{5}+\frac{67067113}{1307639493}a^{3}+\frac{37341311}{145293277}a$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{4865}{932031} a^{15} - \frac{127732}{2796093} a^{13} + \frac{1934282}{2796093} a^{11} - \frac{794750}{2796093} a^{9} + \frac{8894494}{2796093} a^{7} - \frac{1376794}{2796093} a^{5} + \frac{13399556}{2796093} a^{3} + \frac{535048}{310677} a \)  (order $4$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1031923}{1307639493}a^{14}-\frac{1013821}{145293277}a^{12}+\frac{14917509}{145293277}a^{10}-\frac{4044673}{145293277}a^{8}+\frac{59095730}{435879831}a^{6}+\frac{39202953}{145293277}a^{4}+\frac{51218620}{1307639493}a^{2}+\frac{92447095}{145293277}$, $\frac{1931224}{435879831}a^{15}+\frac{3419570}{1307639493}a^{14}-\frac{151834829}{3922918479}a^{13}-\frac{83154781}{3922918479}a^{12}+\frac{2311024903}{3922918479}a^{11}+\frac{434224019}{1307639493}a^{10}-\frac{1005828079}{3922918479}a^{9}+\frac{287184589}{3922918479}a^{8}+\frac{11947113512}{3922918479}a^{7}+\frac{230622028}{145293277}a^{6}-\frac{2990121713}{3922918479}a^{5}+\frac{106663922}{3922918479}a^{4}+\frac{18645921208}{3922918479}a^{3}+\frac{305431571}{145293277}a^{2}+\frac{37451228}{435879831}a-\frac{35371925}{145293277}$, $\frac{1738010}{3922918479}a^{15}-\frac{3556972}{1307639493}a^{14}-\frac{904970}{435879831}a^{13}+\frac{98345039}{3922918479}a^{12}+\frac{5761968}{145293277}a^{11}-\frac{482043250}{1307639493}a^{10}+\frac{34850052}{145293277}a^{9}+\frac{1148432194}{3922918479}a^{8}-\frac{302772398}{1307639493}a^{7}-\frac{199337060}{145293277}a^{6}+\frac{110245882}{145293277}a^{5}+\frac{5151090308}{3922918479}a^{4}-\frac{5261633581}{3922918479}a^{3}-\frac{281887138}{145293277}a^{2}+\frac{494055358}{435879831}a-\frac{52598038}{145293277}$, $\frac{1031923}{1307639493}a^{15}-\frac{3419570}{1307639493}a^{14}-\frac{1013821}{145293277}a^{13}+\frac{83154781}{3922918479}a^{12}+\frac{14917509}{145293277}a^{11}-\frac{434224019}{1307639493}a^{10}-\frac{4044673}{145293277}a^{9}-\frac{287184589}{3922918479}a^{8}+\frac{59095730}{435879831}a^{7}-\frac{230622028}{145293277}a^{6}+\frac{39202953}{145293277}a^{5}-\frac{106663922}{3922918479}a^{4}+\frac{51218620}{1307639493}a^{3}-\frac{305431571}{145293277}a^{2}+\frac{237740372}{145293277}a-\frac{109921352}{145293277}$, $\frac{3238703}{3922918479}a^{14}-\frac{19623467}{1307639493}a^{12}+\frac{656411290}{3922918479}a^{10}-\frac{1278239339}{1307639493}a^{8}-\frac{1840576726}{3922918479}a^{6}-\frac{482486217}{145293277}a^{4}-\frac{173018284}{435879831}a^{2}-\frac{58430303}{145293277}$, $\frac{1283305}{435879831}a^{15}+\frac{5328737}{3922918479}a^{14}-\frac{112449998}{3922918479}a^{13}-\frac{3613120}{435879831}a^{12}+\frac{1611020480}{3922918479}a^{11}+\frac{626706349}{3922918479}a^{10}-\frac{2014940845}{3922918479}a^{9}+\frac{404761658}{1307639493}a^{8}+\frac{5531010331}{3922918479}a^{7}+\frac{8107767782}{3922918479}a^{6}-\frac{13162398038}{3922918479}a^{5}+\frac{2852647028}{1307639493}a^{4}+\frac{2615010893}{3922918479}a^{3}+\frac{2340633994}{435879831}a^{2}-\frac{567579856}{145293277}a+\frac{267423721}{145293277}$, $\frac{926612}{3922918479}a^{15}-\frac{26865040}{1307639493}a^{14}+\frac{7694780}{1307639493}a^{13}+\frac{725412587}{3922918479}a^{12}-\frac{167813932}{3922918479}a^{11}-\frac{1212449762}{435879831}a^{10}+\frac{1438756037}{1307639493}a^{9}+\frac{7521523282}{3922918479}a^{8}-\frac{4204198517}{3922918479}a^{7}-\frac{19097630818}{1307639493}a^{6}+\frac{1133138443}{145293277}a^{5}+\frac{20263312895}{3922918479}a^{4}-\frac{22338399449}{3922918479}a^{3}-\frac{29369048117}{1307639493}a^{2}+\frac{6947035520}{435879831}a-\frac{959593900}{145293277}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 213280.05844 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 213280.05844 \cdot 4}{4\cdot\sqrt{596430979135513600000000}}\cr\approx \mathstrut & 0.67082478701 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 9*x^14 + 135*x^12 - 90*x^10 + 663*x^8 - 270*x^6 + 1150*x^4 + 108*x^2 + 81);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-65}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(i, \sqrt{65})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{-5}, \sqrt{13})\), \(\Q(\sqrt{5}, \sqrt{-13})\), \(\Q(\sqrt{5}, \sqrt{13})\), \(\Q(\sqrt{-5}, \sqrt{-13})\), 4.0.2197.1, 4.4.35152.1, 4.4.878800.1, 4.0.54925.1, 8.0.4569760000.1, 8.0.1235663104.1, 8.0.772289440000.3, 8.0.772289440000.2, 8.0.772289440000.1, 8.0.3016755625.1, 8.8.772289440000.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{8}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ R ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{8}$ ${\href{/padicField/29.1.0.1}{1} }^{16}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(5\) Copy content Toggle raw display 5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(13\) Copy content Toggle raw display 13.8.6.1$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24242 x^{4} + 15024 x^{3} + 14408 x^{2} + 86496 x + 254881$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$