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Properties

Label 8.0.183178223892225.1
Degree $8$
Signature $[0, 4]$
Discriminant $1.832\times 10^{14}$
Root discriminant \(60.65\)
Ramified primes $3,5,67$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_4\times C_2$ (as 8T9)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 - 21*x^6 + 72*x^5 + 548*x^4 - 5112*x^3 + 19563*x^2 - 39405*x + 34225)
 
Copy content gp:K = bnfinit(y^8 - 3*y^7 - 21*y^6 + 72*y^5 + 548*y^4 - 5112*y^3 + 19563*y^2 - 39405*y + 34225, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 3*x^7 - 21*x^6 + 72*x^5 + 548*x^4 - 5112*x^3 + 19563*x^2 - 39405*x + 34225);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 3*x^7 - 21*x^6 + 72*x^5 + 548*x^4 - 5112*x^3 + 19563*x^2 - 39405*x + 34225)
 

\( x^{8} - 3x^{7} - 21x^{6} + 72x^{5} + 548x^{4} - 5112x^{3} + 19563x^{2} - 39405x + 34225 \) 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:  $8$
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:  $[0, 4]$
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:   \(183178223892225\) \(\medspace = 3^{4}\cdot 5^{2}\cdot 67^{6}\) 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:  \(60.65\)
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:  $3^{1/2}5^{1/2}67^{3/4}\approx 90.69883977059361$
Ramified primes:   \(3\), \(5\), \(67\) 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:  \(\Q(\sqrt{-3}, \sqrt{-67})\)

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{10}a^{5}-\frac{1}{5}a^{4}+\frac{1}{10}a^{2}+\frac{2}{5}a$, $\frac{1}{20}a^{6}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{1}{4}$, $\frac{1}{101259380}a^{7}-\frac{2002813}{101259380}a^{6}+\frac{66493}{25314845}a^{5}+\frac{3491264}{25314845}a^{4}+\frac{1459972}{25314845}a^{3}+\frac{10467724}{25314845}a^{2}-\frac{7103225}{20251876}a-\frac{169915}{547348}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

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:  Trivial group, which has order $1$ (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:  $3$
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:   \( -\frac{32111}{50629690} a^{7} - \frac{70959}{101259380} a^{6} + \frac{601811}{50629690} a^{5} + \frac{314083}{50629690} a^{4} - \frac{17586653}{50629690} a^{3} + \frac{91776657}{50629690} a^{2} - \frac{106718459}{25314845} a + \frac{2403957}{547348} \)  (order $6$) 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:   $\frac{270871}{101259380}a^{7}-\frac{442201}{25314845}a^{6}-\frac{973629}{50629690}a^{5}+\frac{3151631}{5062969}a^{4}-\frac{85440482}{25314845}a^{3}+\frac{99632575}{10125938}a^{2}-\frac{1619284923}{101259380}a+\frac{1490744}{136837}$, $\frac{18634449}{50629690}a^{7}-\frac{45951393}{101259380}a^{6}-\frac{480926107}{50629690}a^{5}+\frac{375069277}{50629690}a^{4}+\frac{2355070481}{10125938}a^{3}-\frac{71568974553}{50629690}a^{2}+\frac{105003237662}{25314845}a-\frac{2625967145}{547348}$, $\frac{17\cdots 71}{101259380}a^{7}-\frac{43\cdots 71}{50629690}a^{6}-\frac{19\cdots 29}{50629690}a^{5}+\frac{54\cdots 18}{25314845}a^{4}+\frac{25\cdots 41}{25314845}a^{3}-\frac{31\cdots 09}{50629690}a^{2}+\frac{17\cdots 33}{101259380}a-\frac{60\cdots 85}{273674}$ 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:  \( 41731.4122741 \) (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)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 41731.4122741 \cdot 1}{6\cdot\sqrt{183178223892225}}\cr\approx \mathstrut & 0.800929672508 \end{aligned}\] (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^8 - 3*x^7 - 21*x^6 + 72*x^5 + 548*x^4 - 5112*x^3 + 19563*x^2 - 39405*x + 34225) 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^8 - 3*x^7 - 21*x^6 + 72*x^5 + 548*x^4 - 5112*x^3 + 19563*x^2 - 39405*x + 34225, 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^8 - 3*x^7 - 21*x^6 + 72*x^5 + 548*x^4 - 5112*x^3 + 19563*x^2 - 39405*x + 34225); 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^8 - 3*x^7 - 21*x^6 + 72*x^5 + 548*x^4 - 5112*x^3 + 19563*x^2 - 39405*x + 34225); 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

$C_2\times D_4$ (as 8T9):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 16
The 10 conjugacy class representatives for $D_4\times C_2$
Character table for $D_4\times C_2$

Intermediate fields

\(\Q(\sqrt{201}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-67}) \), 4.2.13534335.1, 4.2.13534335.2, \(\Q(\sqrt{-3}, \sqrt{-67})\)

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

Galois closure: deg 16
Degree 8 siblings: 8.0.508828399700625.1, 8.0.4579455597305625.15, 8.4.4579455597305625.1
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 ${\href{/padicField/2.2.0.1}{2} }^{4}$ R R ${\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(3\) Copy content Toggle raw display 3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.1.0a1.1$x^{2} + 4 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(67\) Copy content Toggle raw display 67.1.4.3a1.1$x^{4} + 67$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$
67.1.4.3a1.1$x^{4} + 67$$4$$1$$3$$D_{4}$$$[\ ]_{4}^{2}$$

Spectrum of ring of integers

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