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Properties

Label 16.0.605...000.3
Degree $16$
Signature $[0, 8]$
Discriminant $6.052\times 10^{23}$
Root discriminant \(30.65\)
Ramified primes $2,3,5,7$
Class number $32$ (GRH)
Class group [4, 8] (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 29*x^14 + 336*x^12 + 1979*x^10 + 6231*x^8 + 10055*x^6 + 7040*x^4 + 1125*x^2 + 25)
 
gp: K = bnfinit(y^16 + 29*y^14 + 336*y^12 + 1979*y^10 + 6231*y^8 + 10055*y^6 + 7040*y^4 + 1125*y^2 + 25, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 29*x^14 + 336*x^12 + 1979*x^10 + 6231*x^8 + 10055*x^6 + 7040*x^4 + 1125*x^2 + 25);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 29*x^14 + 336*x^12 + 1979*x^10 + 6231*x^8 + 10055*x^6 + 7040*x^4 + 1125*x^2 + 25)
 

\( x^{16} + 29x^{14} + 336x^{12} + 1979x^{10} + 6231x^{8} + 10055x^{6} + 7040x^{4} + 1125x^{2} + 25 \) 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:   \(605165749776000000000000\) \(\medspace = 2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 7^{8}\) 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:  \(30.65\)
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:  $2\cdot 3^{1/2}5^{3/4}7^{1/2}\approx 30.645530678223075$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) 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:  \(420=2^{2}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{420}(1,·)$, $\chi_{420}(323,·)$, $\chi_{420}(197,·)$, $\chi_{420}(13,·)$, $\chi_{420}(209,·)$, $\chi_{420}(211,·)$, $\chi_{420}(251,·)$, $\chi_{420}(407,·)$, $\chi_{420}(223,·)$, $\chi_{420}(97,·)$, $\chi_{420}(419,·)$, $\chi_{420}(169,·)$, $\chi_{420}(113,·)$, $\chi_{420}(307,·)$, $\chi_{420}(41,·)$, $\chi_{420}(379,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}-\frac{2}{11}a^{8}-\frac{3}{11}a^{6}+\frac{4}{11}a^{4}+\frac{4}{11}a^{2}-\frac{3}{11}$, $\frac{1}{11}a^{11}-\frac{2}{11}a^{9}-\frac{3}{11}a^{7}+\frac{4}{11}a^{5}+\frac{4}{11}a^{3}-\frac{3}{11}a$, $\frac{1}{55}a^{12}-\frac{1}{55}a^{10}+\frac{6}{55}a^{8}-\frac{21}{55}a^{6}-\frac{14}{55}a^{4}-\frac{2}{11}a^{2}-\frac{5}{11}$, $\frac{1}{275}a^{13}-\frac{6}{275}a^{11}+\frac{126}{275}a^{9}-\frac{61}{275}a^{7}+\frac{21}{275}a^{5}-\frac{17}{55}a^{3}+\frac{9}{55}a$, $\frac{1}{24475}a^{14}+\frac{134}{24475}a^{12}+\frac{611}{24475}a^{10}-\frac{2396}{24475}a^{8}+\frac{7856}{24475}a^{6}+\frac{201}{4895}a^{4}+\frac{174}{4895}a^{2}-\frac{39}{979}$, $\frac{1}{24475}a^{15}-\frac{4}{2225}a^{13}-\frac{546}{24475}a^{11}+\frac{4101}{24475}a^{9}+\frac{914}{24475}a^{7}-\frac{11633}{24475}a^{5}+\frac{284}{979}a^{3}-\frac{42}{445}a$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{4}\times C_{8}$, which has order $32$ (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:   \( \frac{1}{275} a^{15} + \frac{6}{55} a^{13} + \frac{72}{55} a^{11} + 8 a^{9} + \frac{288}{11} a^{7} + \frac{12096}{275} a^{5} + \frac{1792}{55} a^{3} + \frac{384}{55} a \)  (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:   $\frac{9}{4895}a^{14}+\frac{316}{4895}a^{12}+\frac{4164}{4895}a^{10}+\frac{26496}{4895}a^{8}+\frac{86279}{4895}a^{6}+\frac{27975}{979}a^{4}+\frac{19188}{979}a^{2}+\frac{1093}{979}$, $\frac{321}{24475}a^{14}+\frac{8304}{24475}a^{12}+\frac{83991}{24475}a^{10}+\frac{417699}{24475}a^{8}+\frac{1047936}{24475}a^{6}+\frac{240029}{4895}a^{4}+\frac{91009}{4895}a^{2}+\frac{1632}{979}$, $\frac{12}{24475}a^{14}+\frac{273}{24475}a^{12}+\frac{1992}{24475}a^{10}+\frac{1063}{24475}a^{8}-\frac{53468}{24475}a^{6}-\frac{9628}{979}a^{4}-\frac{64217}{4895}a^{2}-\frac{3227}{979}$, $\frac{23}{4895}a^{15}+\frac{3662}{24475}a^{13}+\frac{47303}{24475}a^{11}+\frac{315687}{24475}a^{9}+\frac{1141693}{24475}a^{7}+\frac{192172}{2225}a^{5}+\frac{321186}{4895}a^{3}+\frac{35603}{4895}a$, $\frac{19}{4895}a^{14}+\frac{499}{4895}a^{12}+\frac{5201}{4895}a^{10}+\frac{27634}{4895}a^{8}+\frac{80556}{4895}a^{6}+\frac{126518}{4895}a^{4}+\frac{16300}{979}a^{2}+\frac{834}{979}$, $\frac{658}{24475}a^{15}+\frac{3786}{4895}a^{13}+\frac{43348}{4895}a^{11}+\frac{250673}{4895}a^{9}+\frac{765657}{4895}a^{7}+\frac{5835483}{24475}a^{5}+\frac{708211}{4895}a^{3}+\frac{29932}{4895}a$, $\frac{113}{24475}a^{15}+\frac{2593}{24475}a^{13}+\frac{21962}{24475}a^{11}+\frac{81603}{24475}a^{9}+\frac{111292}{24475}a^{7}-\frac{27589}{24475}a^{5}-\frac{4309}{979}a^{3}+\frac{2084}{4895}a$ 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:  \( 14409.2792961 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 14409.2792961 \cdot 32}{4\cdot\sqrt{605165749776000000000000}}\cr\approx \mathstrut & 0.359943241151 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 29*x^14 + 336*x^12 + 1979*x^10 + 6231*x^8 + 10055*x^6 + 7040*x^4 + 1125*x^2 + 25)
 
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 + 29*x^14 + 336*x^12 + 1979*x^10 + 6231*x^8 + 10055*x^6 + 7040*x^4 + 1125*x^2 + 25, 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 + 29*x^14 + 336*x^12 + 1979*x^10 + 6231*x^8 + 10055*x^6 + 7040*x^4 + 1125*x^2 + 25);
 
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 + 29*x^14 + 336*x^12 + 1979*x^10 + 6231*x^8 + 10055*x^6 + 7040*x^4 + 1125*x^2 + 25);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-105}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(i, \sqrt{105})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-5}, \sqrt{-21})\), \(\Q(\sqrt{5}, \sqrt{-21})\), \(\Q(\sqrt{-5}, \sqrt{21})\), 4.0.98000.1, 4.4.6125.1, 4.0.18000.1, \(\Q(\zeta_{15})^+\), 8.0.31116960000.8, 8.0.9604000000.1, 8.0.324000000.1, 8.0.777924000000.1, 8.8.3038765625.1, 8.0.777924000000.7, 8.0.777924000000.2

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 R R R ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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}$
\(3\) Copy content Toggle raw display 3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$
\(7\) Copy content Toggle raw display 7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$