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 - 216*x^10 - 5994*x^8 - 23328*x^6 - 29160*x^4 - 17496*x^2 - 6561)
 
Copy content gp:K = bnfinit(y^16 - 216*y^10 - 5994*y^8 - 23328*y^6 - 29160*y^4 - 17496*y^2 - 6561, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 216*x^10 - 5994*x^8 - 23328*x^6 - 29160*x^4 - 17496*x^2 - 6561);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 216*x^10 - 5994*x^8 - 23328*x^6 - 29160*x^4 - 17496*x^2 - 6561)
 

\( x^{16} - 216x^{10} - 5994x^{8} - 23328x^{6} - 29160x^{4} - 17496x^{2} - 6561 \) 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:  $(2, 7)$
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:   \(-123933785976430968988237824\) \(\medspace = -\,2^{74}\cdot 3^{8}\) 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:  \(42.74\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{5525/1024}3^{1/2}\approx 72.90718342731029$
Ramified primes:   \(2\), \(3\) 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:  \(\Q(\sqrt{-1}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{135784269}a^{14}-\frac{2164}{5029047}a^{12}+\frac{25093}{15087141}a^{10}-\frac{24499}{5029047}a^{8}+\frac{1285}{558783}a^{6}-\frac{1877}{62087}a^{4}+\frac{3005}{62087}a^{2}+\frac{5888}{62087}$, $\frac{1}{135784269}a^{15}-\frac{2164}{5029047}a^{13}+\frac{25093}{15087141}a^{11}-\frac{24499}{5029047}a^{9}+\frac{1285}{558783}a^{7}-\frac{1877}{62087}a^{5}+\frac{3005}{62087}a^{3}+\frac{5888}{62087}a$ 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:  Trivial group, which has order $1$ (assuming GRH)
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:  $C_{2}$, which has order $2$ (assuming GRH)
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:  $8$
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:   $\frac{29810}{135784269}a^{14}+\frac{149}{1676349}a^{12}+\frac{1846}{15087141}a^{10}+\frac{234157}{5029047}a^{8}+\frac{2178532}{1676349}a^{6}+\frac{846080}{186261}a^{4}+\frac{720430}{186261}a^{2}+\frac{122843}{62087}$, $\frac{9284}{135784269}a^{14}-\frac{17840}{45261423}a^{12}+\frac{12988}{15087141}a^{10}-\frac{86122}{5029047}a^{8}-\frac{177025}{558783}a^{6}+\frac{307585}{558783}a^{4}+\frac{498680}{186261}a^{2}+\frac{89719}{62087}$, $\frac{1685}{15087141}a^{14}+\frac{5681}{45261423}a^{12}-\frac{458}{1676349}a^{10}+\frac{40967}{1676349}a^{8}+\frac{1080358}{1676349}a^{6}+\frac{1066862}{558783}a^{4}+\frac{125207}{62087}a^{2}+\frac{51673}{62087}$, $\frac{6499}{45261423}a^{14}+\frac{520}{45261423}a^{12}-\frac{7339}{15087141}a^{10}-\frac{145886}{5029047}a^{8}-\frac{1454423}{1676349}a^{6}-\frac{1853896}{558783}a^{4}-\frac{375364}{186261}a^{2}-\frac{62614}{62087}$, $\frac{39094}{135784269}a^{14}+\frac{21863}{45261423}a^{12}-\frac{1238}{1676349}a^{10}+\frac{320279}{5029047}a^{8}+\frac{2709607}{1676349}a^{6}+\frac{2230655}{558783}a^{4}+\frac{221750}{186261}a^{2}-\frac{28963}{62087}$, $\frac{357592}{135784269}a^{15}+\frac{66539}{135784269}a^{14}+\frac{54334}{15087141}a^{13}+\frac{152561}{45261423}a^{12}+\frac{253880}{15087141}a^{11}+\frac{826654}{15087141}a^{10}+\frac{3709667}{5029047}a^{9}+\frac{2190356}{5029047}a^{8}+\frac{28063825}{1676349}a^{7}+\frac{3130798}{1676349}a^{6}+\frac{30097099}{558783}a^{5}+\frac{7181813}{558783}a^{4}+\frac{6570556}{186261}a^{3}+\frac{6669887}{186261}a^{2}-\frac{317787}{62087}a+\frac{1564837}{62087}$, $\frac{16658}{45261423}a^{15}-\frac{1240201}{45261423}a^{14}-\frac{5197496}{45261423}a^{13}-\frac{3772850}{15087141}a^{12}-\frac{895661}{15087141}a^{11}+\frac{29180929}{15087141}a^{10}+\frac{14783197}{1676349}a^{9}+\frac{131572984}{5029047}a^{8}+\frac{61534408}{1676349}a^{7}+\frac{138557750}{1676349}a^{6}+\frac{3030004}{62087}a^{5}+\frac{53753633}{558783}a^{4}+\frac{1755019}{62087}a^{3}+\frac{9647450}{186261}a^{2}+\frac{480077}{62087}a+\frac{1108356}{62087}$, $\frac{287283007}{135784269}a^{15}-\frac{4920500}{15087141}a^{14}+\frac{50836999}{45261423}a^{13}+\frac{17821903}{15087141}a^{12}+\frac{4032131}{15087141}a^{11}-\frac{8201029}{15087141}a^{10}+\frac{2295000734}{5029047}a^{9}+\frac{39256415}{558783}a^{8}+\frac{20853161987}{1676349}a^{7}+\frac{316626005}{186261}a^{6}+\frac{23785910357}{558783}a^{5}+\frac{121110698}{186261}a^{4}+\frac{6338108044}{186261}a^{3}-\frac{2733336106}{186261}a^{2}+\frac{115392273}{62087}a-\frac{915214045}{62087}$ Copy content Toggle raw display (assuming GRH)
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:  \( 9269920.160334708 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 1 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{7}\cdot 9269920.160334708 \cdot 1}{2\cdot\sqrt{123933785976430968988237824}}\cr\approx \mathstrut & 0.643828517729542 \end{aligned}\] (assuming GRH)

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 - 216*x^10 - 5994*x^8 - 23328*x^6 - 29160*x^4 - 17496*x^2 - 6561) 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 - 216*x^10 - 5994*x^8 - 23328*x^6 - 29160*x^4 - 17496*x^2 - 6561, 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 - 216*x^10 - 5994*x^8 - 23328*x^6 - 29160*x^4 - 17496*x^2 - 6561); 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 = polynomial_ring(QQ); K, a = number_field(x^16 - 216*x^10 - 5994*x^8 - 23328*x^6 - 29160*x^4 - 17496*x^2 - 6561); 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

$C_2^7.D_8$ (as 16T1476):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 2048
The 47 conjugacy class representatives for $C_2^7.D_8$
Character table for $C_2^7.D_8$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{1 + \sqrt{2}})\), 8.4.536870912.1

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: 16.6.123933785976430968988237824.3, 16.2.123933785976430968988237824.13, 16.2.123933785976430968988237824.10, 16.6.123933785976430968988237824.5, 16.2.123933785976430968988237824.21, 16.6.123933785976430968988237824.7, 16.6.123933785976430968988237824.8, 16.4.61966892988215484494118912.48, 16.4.61966892988215484494118912.54, 16.4.61966892988215484494118912.53, 16.0.61966892988215484494118912.363, 16.0.61966892988215484494118912.364, 16.4.61966892988215484494118912.57, 16.0.61966892988215484494118912.365, 16.0.61966892988215484494118912.373
Degree 32 siblings: deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32, deg 32
Minimal sibling: 16.4.61966892988215484494118912.48

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 ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$ $16$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{6}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ $16$

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 2.1.16.74a1.11$x^{16} + 16 x^{13} + 20 x^{12} + 16 x^{11} + 16 x^{10} + 32 x^{9} + 8 x^{8} + 32 x^{7} + 16 x^{6} + 32 x^{5} + 8 x^{4} + 32 x^{3} + 16 x^{2} + 32 x + 2$$16$$1$$74$16T1476$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{9}{2}, \frac{19}{4}, \frac{39}{8}, \frac{21}{4}, \frac{43}{8}, \frac{45}{8}]$$
\(3\) Copy content Toggle raw display 3.8.2.8a1.1$x^{16} + 4 x^{13} + 2 x^{12} + 8 x^{10} + 8 x^{9} + 5 x^{8} + 8 x^{7} + 12 x^{6} + 12 x^{5} + 8 x^{4} + 8 x^{3} + 12 x^{2} + 11 x + 4$$2$$8$$8$$C_{16}$$$[\ ]_{2}^{8}$$

Spectrum of ring of integers

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