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Properties

Label 16.0.5821863490955437.1
Degree $16$
Signature $[0, 8]$
Discriminant $5.822\times 10^{15}$
Root discriminant \(9.67\)
Ramified primes $19,101,277,2389$
Class number $1$
Class group trivial
Galois group $C_2^8.S_8$ (as 16T1948)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 17*x^13 + 23*x^12 - 27*x^11 + 30*x^10 - 30*x^9 + 25*x^8 - 14*x^7 + 6*x^6 - 7*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^16 - 4*y^15 + 10*y^14 - 17*y^13 + 23*y^12 - 27*y^11 + 30*y^10 - 30*y^9 + 25*y^8 - 14*y^7 + 6*y^6 - 7*y^5 + 15*y^4 - 20*y^3 + 15*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 10*x^14 - 17*x^13 + 23*x^12 - 27*x^11 + 30*x^10 - 30*x^9 + 25*x^8 - 14*x^7 + 6*x^6 - 7*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 4*x^15 + 10*x^14 - 17*x^13 + 23*x^12 - 27*x^11 + 30*x^10 - 30*x^9 + 25*x^8 - 14*x^7 + 6*x^6 - 7*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1)
 

\( x^{16} - 4 x^{15} + 10 x^{14} - 17 x^{13} + 23 x^{12} - 27 x^{11} + 30 x^{10} - 30 x^{9} + 25 x^{8} + \cdots + 1 \) 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:   \(5821863490955437\) \(\medspace = 19^{2}\cdot 101^{2}\cdot 277\cdot 2389^{2}\) 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:  \(9.67\)
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:  $19^{1/2}101^{1/2}277^{1/2}2389^{1/2}\approx 35635.71252269274$
Ramified primes:   \(19\), \(101\), \(277\), \(2389\) 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:  \(\Q(\sqrt{277}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{37}a^{15}-\frac{15}{37}a^{14}-\frac{10}{37}a^{13}-\frac{18}{37}a^{12}-\frac{1}{37}a^{11}-\frac{16}{37}a^{10}-\frac{16}{37}a^{9}-\frac{2}{37}a^{8}+\frac{10}{37}a^{7}-\frac{13}{37}a^{6}+\frac{1}{37}a^{5}-\frac{18}{37}a^{4}-\frac{9}{37}a^{3}+\frac{5}{37}a^{2}-\frac{3}{37}a-\frac{10}{37}$ 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

Trivial group, which has order $1$

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:   \( -1 \)  (order $2$) 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:   $\frac{378}{37}a^{15}-\frac{1267}{37}a^{14}+\frac{2954}{37}a^{13}-\frac{4510}{37}a^{12}+\frac{5764}{37}a^{11}-\frac{6455}{37}a^{10}+\frac{7124}{37}a^{9}-\frac{6676}{37}a^{8}+\frac{5075}{37}a^{7}-\frac{1954}{37}a^{6}+\frac{933}{37}a^{5}-\frac{1994}{37}a^{4}+\frac{4331}{37}a^{3}-\frac{4733}{37}a^{2}+\frac{2603}{37}a-\frac{561}{37}$, $\frac{480}{37}a^{15}-\frac{1613}{37}a^{14}+\frac{3784}{37}a^{13}-\frac{5791}{37}a^{12}+\frac{7438}{37}a^{11}-\frac{8346}{37}a^{10}+\frac{9229}{37}a^{9}-\frac{8693}{37}a^{8}+\frac{6650}{37}a^{7}-\frac{2651}{37}a^{6}+\frac{1294}{37}a^{5}-\frac{2572}{37}a^{4}+\frac{5596}{37}a^{3}-\frac{6110}{37}a^{2}+\frac{3407}{37}a-\frac{804}{37}$, $a$, $\frac{344}{37}a^{15}-\frac{1127}{37}a^{14}+\frac{2628}{37}a^{13}-\frac{3972}{37}a^{12}+\frac{5095}{37}a^{11}-\frac{5689}{37}a^{10}+\frac{6299}{37}a^{9}-\frac{5868}{37}a^{8}+\frac{4476}{37}a^{7}-\frac{1697}{37}a^{6}+\frac{899}{37}a^{5}-\frac{1752}{37}a^{4}+\frac{3860}{37}a^{3}-\frac{4126}{37}a^{2}+\frac{2261}{37}a-\frac{517}{37}$, $\frac{344}{37}a^{15}-\frac{1127}{37}a^{14}+\frac{2591}{37}a^{13}-\frac{3898}{37}a^{12}+\frac{4947}{37}a^{11}-\frac{5541}{37}a^{10}+\frac{6114}{37}a^{9}-\frac{5683}{37}a^{8}+\frac{4254}{37}a^{7}-\frac{1586}{37}a^{6}+\frac{825}{37}a^{5}-\frac{1826}{37}a^{4}+\frac{3749}{37}a^{3}-\frac{4015}{37}a^{2}+\frac{2076}{37}a-\frac{443}{37}$, $\frac{405}{37}a^{15}-\frac{1413}{37}a^{14}+\frac{3313}{37}a^{13}-\frac{5144}{37}a^{12}+\frac{6588}{37}a^{11}-\frac{7442}{37}a^{10}+\frac{8209}{37}a^{9}-\frac{7803}{37}a^{8}+\frac{5974}{37}a^{7}-\frac{2490}{37}a^{6}+\frac{1108}{37}a^{5}-\frac{2295}{37}a^{4}+\frac{4902}{37}a^{3}-\frac{5523}{37}a^{2}+\frac{3114}{37}a-\frac{720}{37}$, $\frac{324}{37}a^{15}-\frac{1123}{37}a^{14}+\frac{2643}{37}a^{13}-\frac{4093}{37}a^{12}+\frac{5263}{37}a^{11}-\frac{5924}{37}a^{10}+\frac{6545}{37}a^{9}-\frac{6198}{37}a^{8}+\frac{4757}{37}a^{7}-\frac{1955}{37}a^{6}+\frac{879}{37}a^{5}-\frac{1762}{37}a^{4}+\frac{3892}{37}a^{3}-\frac{4337}{37}a^{2}+\frac{2506}{37}a-\frac{576}{37}$ 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:  \( 10.6211653408 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10.6211653408 \cdot 1}{2\cdot\sqrt{5821863490955437}}\cr\approx \mathstrut & 0.169063599720 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 10*x^14 - 17*x^13 + 23*x^12 - 27*x^11 + 30*x^10 - 30*x^9 + 25*x^8 - 14*x^7 + 6*x^6 - 7*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*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^15 + 10*x^14 - 17*x^13 + 23*x^12 - 27*x^11 + 30*x^10 - 30*x^9 + 25*x^8 - 14*x^7 + 6*x^6 - 7*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*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^15 + 10*x^14 - 17*x^13 + 23*x^12 - 27*x^11 + 30*x^10 - 30*x^9 + 25*x^8 - 14*x^7 + 6*x^6 - 7*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*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^15 + 10*x^14 - 17*x^13 + 23*x^12 - 27*x^11 + 30*x^10 - 30*x^9 + 25*x^8 - 14*x^7 + 6*x^6 - 7*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*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

$C_2^8.S_8$ (as 16T1948):

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 non-solvable group of order 10321920
The 185 conjugacy class representatives for $C_2^8.S_8$
Character table for $C_2^8.S_8$

Intermediate fields

8.2.4584491.1

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 16 siblings: data not computed
Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $16$ ${\href{/padicField/3.8.0.1}{8} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ R ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

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
\(19\) Copy content Toggle raw display 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}$
19.10.0.1$x^{10} + 18 x^{5} + 13 x^{4} + 17 x^{3} + 3 x^{2} + 4 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(101\) Copy content Toggle raw display $\Q_{101}$$x + 99$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 99$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.4.2.1$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.6.0.1$x^{6} + 2 x^{4} + 90 x^{3} + 20 x^{2} + 67 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(277\) Copy content Toggle raw display $\Q_{277}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{277}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(2389\) Copy content Toggle raw display $\Q_{2389}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2389}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2389}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2389}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$2$$2$$2$