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Properties

Label 16.0.358...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $3.581\times 10^{21}$
Root discriminant \(22.24\)
Ramified primes $5,19,103$
Class number $5$
Class group [5]
Galois group $C_4\times S_4$ (as 16T181)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1)
 
gp: K = bnfinit(y^16 + 4*y^14 - y^13 + 15*y^12 + 12*y^11 + 57*y^10 + 34*y^9 + 201*y^8 + 67*y^7 + 74*y^6 + 33*y^5 + 28*y^4 - 9*y^3 + 5*y^2 - y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1)
 

\( x^{16} + 4 x^{14} - x^{13} + 15 x^{12} + 12 x^{11} + 57 x^{10} + 34 x^{9} + 201 x^{8} + 67 x^{7} + \cdots + 1 \) 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:   \(3580992031933837890625\) \(\medspace = 5^{12}\cdot 19^{4}\cdot 103^{4}\) 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:  \(22.24\)
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:  $5^{3/4}19^{1/2}103^{1/2}\approx 147.91864371956444$
Ramified primes:   \(5\), \(19\), \(103\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{42142}a^{13}+\frac{25}{21071}a^{12}+\frac{3892}{21071}a^{11}+\frac{199}{42142}a^{10}+\frac{5007}{21071}a^{9}+\frac{18579}{42142}a^{8}-\frac{7275}{21071}a^{7}+\frac{5816}{21071}a^{6}-\frac{2509}{42142}a^{5}-\frac{10357}{21071}a^{4}+\frac{2143}{42142}a^{3}+\frac{6432}{21071}a^{2}-\frac{758}{21071}a-\frac{14257}{42142}$, $\frac{1}{42142}a^{14}+\frac{2642}{21071}a^{12}-\frac{9723}{42142}a^{11}+\frac{32}{21071}a^{10}-\frac{18559}{42142}a^{9}-\frac{8188}{21071}a^{8}-\frac{9712}{21071}a^{7}+\frac{5879}{42142}a^{6}+\frac{10226}{21071}a^{5}-\frac{15707}{42142}a^{4}-\frac{5001}{21071}a^{3}-\frac{6293}{21071}a^{2}+\frac{19401}{42142}a-\frac{1782}{21071}$, $\frac{1}{42142}a^{15}+\frac{2273}{21071}a^{10}-\frac{5905}{21071}a^{5}+\frac{5163}{42142}$ 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:  $2$

Class group and class number

$C_{5}$, which has order $5$

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

Relative class number: $5$

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:   \( \frac{3324}{21071} a^{15} - \frac{831}{21071} a^{14} + \frac{12465}{21071} a^{13} - \frac{13161}{42142} a^{12} + \frac{2493}{1109} a^{11} + \frac{28254}{21071} a^{10} + \frac{167031}{21071} a^{9} + \frac{55677}{21071} a^{8} + \frac{1187657}{42142} a^{7} + \frac{27423}{21071} a^{6} + \frac{23268}{21071} a^{5} - \frac{7479}{21071} a^{4} + \frac{4155}{21071} a^{3} - \frac{259523}{42142} a^{2} + \frac{831}{21071} a \)  (order $10$) 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:   $\frac{7407}{42142}a^{15}+\frac{7407}{42142}a^{14}+\frac{29749}{42142}a^{13}+\frac{22221}{42142}a^{12}+\frac{51849}{21071}a^{11}+\frac{199989}{42142}a^{10}+\frac{511083}{42142}a^{9}+\frac{676257}{42142}a^{8}+\frac{1740645}{42142}a^{7}+\frac{992538}{21071}a^{6}+\frac{1044387}{42142}a^{5}+\frac{792549}{42142}a^{4}+\frac{371361}{42142}a^{3}+\frac{7407}{2218}a^{2}-\frac{14814}{21071}a-\frac{6257}{21071}$, $a$, $\frac{5084}{21071}a^{14}-\frac{1271}{21071}a^{13}+\frac{19065}{21071}a^{12}-\frac{10331}{21071}a^{11}+\frac{3813}{1109}a^{10}+\frac{43214}{21071}a^{9}+\frac{255471}{21071}a^{8}+\frac{85157}{21071}a^{7}+\frac{902710}{21071}a^{6}+\frac{41943}{21071}a^{5}+\frac{35588}{21071}a^{4}-\frac{11439}{21071}a^{3}+\frac{6355}{21071}a^{2}-\frac{31434}{21071}a+\frac{1271}{21071}$, $\frac{242}{21071}a^{14}-\frac{121}{42142}a^{13}+\frac{1815}{42142}a^{12}-\frac{384}{21071}a^{11}+\frac{363}{2218}a^{10}+\frac{2057}{21071}a^{9}+\frac{24321}{42142}a^{8}+\frac{8107}{42142}a^{7}+\frac{44710}{21071}a^{6}+\frac{3993}{42142}a^{5}+\frac{1694}{21071}a^{4}-\frac{1089}{42142}a^{3}+\frac{605}{42142}a^{2}-\frac{59570}{21071}a+\frac{121}{42142}$, $\frac{14055}{42142}a^{15}-\frac{7705}{42142}a^{14}+\frac{56099}{42142}a^{13}-\frac{43683}{42142}a^{12}+\frac{109116}{21071}a^{11}+\frac{57555}{42142}a^{10}+\frac{706087}{42142}a^{9}+\frac{53451}{42142}a^{8}+\frac{2573217}{42142}a^{7}-\frac{273561}{21071}a^{6}+\frac{543801}{42142}a^{5}+\frac{105859}{42142}a^{4}+\frac{229575}{42142}a^{3}-\frac{333891}{42142}a^{2}+\frac{68469}{21071}a-\frac{4474}{21071}$, $\frac{20219}{42142}a^{15}-\frac{980}{21071}a^{14}+\frac{79335}{42142}a^{13}-\frac{696}{1109}a^{12}+\frac{149331}{21071}a^{11}+\frac{110527}{21071}a^{10}+\frac{551246}{21071}a^{9}+\frac{581117}{42142}a^{8}+\frac{1961646}{21071}a^{7}+\frac{496269}{21071}a^{6}+\frac{541609}{21071}a^{5}+\frac{360287}{21071}a^{4}+\frac{401245}{42142}a^{3}-\frac{112368}{21071}a^{2}+\frac{56979}{21071}a-\frac{14055}{42142}$, $\frac{15177}{42142}a^{15}+\frac{5745}{42142}a^{14}+\frac{54679}{42142}a^{13}+\frac{4530}{21071}a^{12}+\frac{99216}{21071}a^{11}+\frac{139499}{21071}a^{10}+\frac{845145}{42142}a^{9}+\frac{787611}{42142}a^{8}+\frac{1464151}{21071}a^{7}+\frac{1019961}{21071}a^{6}+\frac{188674}{21071}a^{5}+\frac{777591}{42142}a^{4}+\frac{379671}{42142}a^{3}-\frac{59395}{21071}a^{2}-\frac{13983}{21071}a+\frac{70131}{42142}$ Copy content Toggle raw display
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:  \( 10491.0986114 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10491.0986114 \cdot 5}{10\cdot\sqrt{3580992031933837890625}}\cr\approx \mathstrut & 0.212925778153 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1)
 
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 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1, 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 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1);
 
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 + 4*x^14 - x^13 + 15*x^12 + 12*x^11 + 57*x^10 + 34*x^9 + 201*x^8 + 67*x^7 + 74*x^6 + 33*x^5 + 28*x^4 - 9*x^3 + 5*x^2 - x + 1);
 
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

$C_4\times S_4$ (as 16T181):

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)
 
A solvable group of order 96
The 20 conjugacy class representatives for $C_4\times S_4$
Character table for $C_4\times S_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.1957.1, 8.8.2393655625.1

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]
 

Sibling fields

Degree 12 siblings: data not computed
Degree 16 sibling: data not computed
Degree 24 siblings: data not computed
Degree 32 sibling: data not computed
Minimal sibling: 12.0.28647936255470703125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ R ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.2.0.1}{2} }^{8}$

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
\(5\) Copy content Toggle raw display 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(19\) Copy content Toggle raw display 19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} + 18 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(103\) Copy content Toggle raw display 103.4.0.1$x^{4} + 2 x^{2} + 88 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.4.0.1$x^{4} + 2 x^{2} + 88 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.8.4.1$x^{8} + 416 x^{6} + 176 x^{5} + 64080 x^{4} - 35904 x^{3} + 4330880 x^{2} - 5564416 x + 109124096$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$