↑

Properties

Label 16.0.179...000.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.794\times 10^{22}$
Root discriminant \(24.60\)
Ramified primes $2,3,5,61$
Class number $4$
Class group [4]
Galois group $C_2^5:C_4$ (as 16T227)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 + 49*x^12 + 108*x^10 + 196*x^8 - 450*x^6 + 260*x^4 - 75*x^2 + 25)
 
Copy content gp:K = bnfinit(y^16 + 7*y^14 + 49*y^12 + 108*y^10 + 196*y^8 - 450*y^6 + 260*y^4 - 75*y^2 + 25, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 7*x^14 + 49*x^12 + 108*x^10 + 196*x^8 - 450*x^6 + 260*x^4 - 75*x^2 + 25);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 + 7*x^14 + 49*x^12 + 108*x^10 + 196*x^8 - 450*x^6 + 260*x^4 - 75*x^2 + 25)
 

\( x^{16} + 7x^{14} + 49x^{12} + 108x^{10} + 196x^{8} - 450x^{6} + 260x^{4} - 75x^{2} + 25 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(17944209936000000000000\) \(\medspace = 2^{16}\cdot 3^{4}\cdot 5^{12}\cdot 61^{4}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(24.60\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2\cdot 3^{1/2}5^{3/4}61^{1/2}\approx 90.46551164752914$
Ramified primes:   \(2\), \(3\), \(5\), \(61\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  8.0.14884000000.3

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{10}-\frac{1}{5}a^{8}-\frac{2}{5}a^{6}+\frac{1}{5}a^{4}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{11}-\frac{1}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{5}$, $\frac{1}{1360898275}a^{14}+\frac{131672142}{1360898275}a^{12}+\frac{354612689}{1360898275}a^{10}+\frac{30386508}{123718025}a^{8}+\frac{165495856}{1360898275}a^{6}+\frac{26970484}{272179655}a^{4}+\frac{101191166}{272179655}a^{2}-\frac{10279391}{54435931}$, $\frac{1}{1360898275}a^{15}+\frac{131672142}{1360898275}a^{13}+\frac{354612689}{1360898275}a^{11}+\frac{30386508}{123718025}a^{9}+\frac{165495856}{1360898275}a^{7}+\frac{26970484}{272179655}a^{5}+\frac{101191166}{272179655}a^{3}-\frac{10279391}{54435931}a$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  $C_{4}$, which has order $4$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{4}$, which has order $4$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $7$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{14607633}{1360898275}a^{14}-\frac{98539321}{1360898275}a^{12}-\frac{679524057}{1360898275}a^{10}-\frac{119741504}{123718025}a^{8}-\frac{1943040228}{1360898275}a^{6}+\frac{1743480406}{272179655}a^{4}-\frac{619108153}{272179655}a^{2}+\frac{29350966}{54435931}$, $\frac{2887}{136705}a^{14}+\frac{21266}{136705}a^{12}+\frac{148927}{136705}a^{10}+\frac{362884}{136705}a^{8}+\frac{674743}{136705}a^{6}-\frac{1127703}{136705}a^{4}+\frac{37832}{27341}a^{2}+\frac{15540}{27341}$, $\frac{691851}{54435931}a^{14}+\frac{6869031}{54435931}a^{12}+\frac{49687716}{54435931}a^{10}+\frac{16932716}{4948721}a^{8}+\frac{440256882}{54435931}a^{6}+\frac{311668175}{54435931}a^{4}-\frac{287967138}{54435931}a^{2}+\frac{34800022}{54435931}$, $\frac{496654}{123718025}a^{14}-\frac{3768243}{123718025}a^{12}-\frac{26209656}{123718025}a^{10}-\frac{67881052}{123718025}a^{8}-\frac{129468674}{123718025}a^{6}+\frac{26037029}{24743605}a^{4}-\frac{13807014}{24743605}a^{2}+a-\frac{2823447}{4948721}$, $\frac{16480001}{1360898275}a^{15}-\frac{2887}{136705}a^{14}-\frac{105290507}{1360898275}a^{13}-\frac{21266}{136705}a^{12}-\frac{737567719}{1360898275}a^{11}-\frac{148927}{136705}a^{10}-\frac{117179993}{123718025}a^{9}-\frac{362884}{136705}a^{8}-\frac{2171760076}{1360898275}a^{7}-\frac{674743}{136705}a^{6}+\frac{1870148788}{272179655}a^{5}+\frac{1127703}{136705}a^{4}-\frac{1829285811}{272179655}a^{3}-\frac{37832}{27341}a^{2}+\frac{192174839}{54435931}a-\frac{15540}{27341}$, $\frac{77155424}{1360898275}a^{15}-\frac{38732228}{1360898275}a^{14}-\frac{551027418}{1360898275}a^{13}-\frac{298995731}{1360898275}a^{12}-\frac{3861795981}{1360898275}a^{11}-\frac{2098354327}{1360898275}a^{10}-\frac{809724832}{123718025}a^{9}-\frac{507483794}{123718025}a^{8}-\frac{16576242849}{1360898275}a^{7}-\frac{10808567283}{1360898275}a^{6}+\frac{6363115782}{272179655}a^{5}+\frac{2336410887}{272179655}a^{4}-\frac{3309512124}{272179655}a^{3}+\frac{500936652}{272179655}a^{2}+\frac{214230219}{54435931}a-\frac{30189680}{54435931}$, $\frac{38732228}{1360898275}a^{15}+\frac{2887}{136705}a^{14}+\frac{298995731}{1360898275}a^{13}+\frac{21266}{136705}a^{12}+\frac{2098354327}{1360898275}a^{11}+\frac{148927}{136705}a^{10}+\frac{507483794}{123718025}a^{9}+\frac{362884}{136705}a^{8}+\frac{10808567283}{1360898275}a^{7}+\frac{674743}{136705}a^{6}-\frac{2336410887}{272179655}a^{5}-\frac{1127703}{136705}a^{4}-\frac{500936652}{272179655}a^{3}+\frac{37832}{27341}a^{2}+\frac{30189680}{54435931}a-\frac{11801}{27341}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 10557.253437747004 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10557.253437747004 \cdot 4}{2\cdot\sqrt{17944209936000000000000}}\cr\approx \mathstrut & 0.382875625313931 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 + 7*x^14 + 49*x^12 + 108*x^10 + 196*x^8 - 450*x^6 + 260*x^4 - 75*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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 + 7*x^14 + 49*x^12 + 108*x^10 + 196*x^8 - 450*x^6 + 260*x^4 - 75*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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 + 7*x^14 + 49*x^12 + 108*x^10 + 196*x^8 - 450*x^6 + 260*x^4 - 75*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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 + 7*x^14 + 49*x^12 + 108*x^10 + 196*x^8 - 450*x^6 + 260*x^4 - 75*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

$C_2^5:C_4$ (as 16T227):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^5:C_4$
Character table for $C_2^5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.0.1525.1, 4.0.122000.2, 8.0.26791200000.6, 8.0.104653125.1, 8.0.14884000000.3

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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 R R R ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.8.0.1}{8} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.8.0.1}{8} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{8}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.8.2.16a1.1$x^{16} + 2 x^{12} + 2 x^{11} + 2 x^{10} + 5 x^{8} + 2 x^{7} + 3 x^{6} + 2 x^{5} + 5 x^{4} + 4 x^{3} + 4 x^{2} + 5$$2$$8$$16$$C_8\times C_2$$$[2]^{8}$$
\(3\) Copy content Toggle raw display 3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.1.0a1.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
3.4.2.4a1.2$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(5\) Copy content Toggle raw display 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.4.6a1.2$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$$4$$2$$6$$C_4\times C_2$$$[\ ]_{4}^{2}$$
\(61\) Copy content Toggle raw display 61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.1.0a1.1$x^{2} + 60 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
61.2.2.2a1.2$x^{4} + 120 x^{3} + 3604 x^{2} + 240 x + 65$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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