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Properties

Label 16.0.780...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $7.807\times 10^{19}$
Root discriminant \(17.51\)
Ramified primes $2,5,47$
Class number $2$
Class group [2]
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 + 4*x^12 - 2*x^10 + 4*x^8 + 16*x^6 + 21*x^4 + 12*x^2 + 16)
 
gp: K = bnfinit(y^16 + 4*y^14 + 4*y^12 - 2*y^10 + 4*y^8 + 16*y^6 + 21*y^4 + 12*y^2 + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 4*x^14 + 4*x^12 - 2*x^10 + 4*x^8 + 16*x^6 + 21*x^4 + 12*x^2 + 16);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 4*x^14 + 4*x^12 - 2*x^10 + 4*x^8 + 16*x^6 + 21*x^4 + 12*x^2 + 16)
 

\( x^{16} + 4x^{14} + 4x^{12} - 2x^{10} + 4x^{8} + 16x^{6} + 21x^{4} + 12x^{2} + 16 \) 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:   \(78074896000000000000\) \(\medspace = 2^{16}\cdot 5^{12}\cdot 47^{4}\) 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:  \(17.51\)
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:  $2\cdot 5^{3/4}47^{1/2}\approx 45.84652548285205$
Ramified primes:   \(2\), \(5\), \(47\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}+\frac{1}{8}a^{6}+\frac{1}{8}a^{4}+\frac{1}{8}a^{3}$, $\frac{1}{472}a^{14}+\frac{33}{472}a^{12}-\frac{1}{8}a^{11}-\frac{21}{236}a^{10}-\frac{1}{8}a^{9}+\frac{19}{472}a^{8}+\frac{83}{472}a^{6}+\frac{1}{8}a^{5}-\frac{57}{236}a^{4}+\frac{1}{8}a^{3}-\frac{5}{59}a^{2}+\frac{4}{59}$, $\frac{1}{472}a^{15}-\frac{13}{236}a^{13}-\frac{21}{236}a^{11}-\frac{5}{59}a^{9}-\frac{47}{236}a^{7}-\frac{57}{236}a^{5}-\frac{1}{2}a^{4}-\frac{217}{472}a^{3}-\frac{1}{2}a^{2}-\frac{51}{118}a$ 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:  $2$

Class group and class number

$C_{2}$, which has order $2$

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:   $\frac{5}{118}a^{14}+\frac{35}{236}a^{12}-\frac{7}{236}a^{10}-\frac{105}{236}a^{8}+\frac{63}{236}a^{6}+\frac{217}{236}a^{4}+\frac{13}{236}a^{2}-\frac{38}{59}$, $\frac{2}{59}a^{15}-\frac{1}{59}a^{14}+\frac{7}{59}a^{13}-\frac{7}{118}a^{12}+\frac{9}{118}a^{11}-\frac{9}{236}a^{10}-\frac{25}{236}a^{9}-\frac{17}{236}a^{8}+\frac{37}{118}a^{7}-\frac{24}{59}a^{6}+\frac{75}{118}a^{5}-\frac{75}{236}a^{4}+\frac{93}{236}a^{3}-\frac{17}{236}a^{2}+\frac{5}{59}a-\frac{32}{59}$, $\frac{1}{59}a^{14}+\frac{7}{118}a^{12}+\frac{9}{236}a^{10}-\frac{21}{118}a^{8}-\frac{11}{118}a^{6}+\frac{75}{236}a^{4}-\frac{1}{2}a^{3}+\frac{19}{59}a^{2}-\frac{1}{2}a-\frac{27}{59}$, $\frac{1}{59}a^{14}+\frac{7}{118}a^{12}+\frac{9}{236}a^{10}-\frac{21}{118}a^{8}-\frac{11}{118}a^{6}+\frac{75}{236}a^{4}+\frac{1}{2}a^{3}+\frac{19}{59}a^{2}+\frac{1}{2}a-\frac{27}{59}$, $\frac{2}{59}a^{15}+\frac{1}{59}a^{14}+\frac{7}{59}a^{13}+\frac{7}{118}a^{12}+\frac{9}{118}a^{11}+\frac{9}{236}a^{10}-\frac{25}{236}a^{9}+\frac{17}{236}a^{8}+\frac{37}{118}a^{7}+\frac{24}{59}a^{6}+\frac{75}{118}a^{5}+\frac{75}{236}a^{4}+\frac{93}{236}a^{3}+\frac{17}{236}a^{2}+\frac{5}{59}a+\frac{32}{59}$, $\frac{15}{236}a^{15}+\frac{1}{236}a^{14}+\frac{105}{472}a^{13}+\frac{7}{472}a^{12}+\frac{19}{236}a^{11}-\frac{25}{472}a^{10}-\frac{197}{472}a^{9}-\frac{10}{59}a^{8}+\frac{71}{472}a^{7}-\frac{11}{472}a^{6}+\frac{237}{236}a^{5}+\frac{185}{472}a^{4}+\frac{511}{472}a^{3}-\frac{99}{236}a^{2}-\frac{55}{118}a-\frac{51}{59}$, $\frac{7}{472}a^{15}+\frac{1}{236}a^{14}-\frac{5}{472}a^{13}+\frac{7}{472}a^{12}-\frac{29}{236}a^{11}-\frac{25}{472}a^{10}+\frac{15}{472}a^{9}-\frac{10}{59}a^{8}+\frac{109}{472}a^{7}-\frac{11}{472}a^{6}-\frac{163}{236}a^{5}-\frac{51}{472}a^{4}-\frac{81}{236}a^{3}-\frac{217}{236}a^{2}-\frac{3}{118}a+\frac{8}{59}$ 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:  \( 1300.69001393 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 1300.69001393 \cdot 2}{2\cdot\sqrt{78074896000000000000}}\cr\approx \mathstrut & 0.357566682900 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 + 4*x^14 + 4*x^12 - 2*x^10 + 4*x^8 + 16*x^6 + 21*x^4 + 12*x^2 + 16)
 
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 + 4*x^12 - 2*x^10 + 4*x^8 + 16*x^6 + 21*x^4 + 12*x^2 + 16, 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(Rationals()); K<a> := NumberField(x^16 + 4*x^14 + 4*x^12 - 2*x^10 + 4*x^8 + 16*x^6 + 21*x^4 + 12*x^2 + 16);
 
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 + 4*x^12 - 2*x^10 + 4*x^8 + 16*x^6 + 21*x^4 + 12*x^2 + 16);
 
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.0.23500.1, 8.0.552250000.1

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: 12.4.1561497920000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ R ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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 2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
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 5.16.12.1$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}$
\(47\) Copy content Toggle raw display 47.4.0.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.0.1$x^{4} + 8 x^{2} + 40 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
47.8.4.1$x^{8} + 204 x^{6} + 80 x^{5} + 14080 x^{4} - 6880 x^{3} + 384824 x^{2} - 499680 x + 3453444$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)