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Properties

Label 8.0.1046738867982336.20
Degree $8$
Signature $[0, 4]$
Discriminant $1.047\times 10^{15}$
Root discriminant \(75.42\)
Ramified primes $2,3,79$
Class number $80$
Class group [2, 2, 20]
Galois group $S_4$ (as 8T14)

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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 - 2*x^7 - 41*x^6 - 218*x^5 + 865*x^4 + 5296*x^3 + 36682*x^2 + 71584*x + 99628)
 
Copy content gp:K = bnfinit(y^8 - 2*y^7 - 41*y^6 - 218*y^5 + 865*y^4 + 5296*y^3 + 36682*y^2 + 71584*y + 99628, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 41*x^6 - 218*x^5 + 865*x^4 + 5296*x^3 + 36682*x^2 + 71584*x + 99628);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 - 41*x^6 - 218*x^5 + 865*x^4 + 5296*x^3 + 36682*x^2 + 71584*x + 99628)
 

\( x^{8} - 2x^{7} - 41x^{6} - 218x^{5} + 865x^{4} + 5296x^{3} + 36682x^{2} + 71584x + 99628 \) 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:   \(1046738867982336\) \(\medspace = 2^{12}\cdot 3^{8}\cdot 79^{4}\) 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:  \(75.42\)
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:  $2^{3/2}3^{7/6}79^{1/2}\approx 90.57326071147344$
Ramified primes:   \(2\), \(3\), \(79\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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:  \(\Q(\sqrt{-474}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7}a^{4}-\frac{1}{7}a^{3}+\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{5}-\frac{1}{7}a^{3}+\frac{3}{7}a^{2}+\frac{1}{7}a-\frac{2}{7}$, $\frac{1}{42}a^{6}-\frac{1}{14}a^{4}-\frac{1}{21}a^{3}+\frac{5}{14}a^{2}+\frac{1}{7}a+\frac{2}{21}$, $\frac{1}{451049155662}a^{7}+\frac{1096326965}{150349718554}a^{6}+\frac{2913141393}{150349718554}a^{5}+\frac{14851953835}{451049155662}a^{4}-\frac{39373331857}{150349718554}a^{3}-\frac{2980706563}{21478531222}a^{2}+\frac{3937004}{32217796833}a-\frac{26194091043}{75174859277}$ 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:  $C_{2}\times C_{2}\times C_{20}$, which has order $80$
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:  $C_{2}\times C_{2}\times C_{20}$, which has order $80$
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:   \( -1 \)  (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:   $\frac{31075}{10739265611}a^{7}+\frac{39602609}{225524577831}a^{6}-\frac{75391023}{75174859277}a^{5}-\frac{537267533}{75174859277}a^{4}-\frac{3260231725}{225524577831}a^{3}+\frac{18374973662}{75174859277}a^{2}+\frac{47491771968}{75174859277}a+\frac{33380106971}{32217796833}$, $\frac{4208381}{150349718554}a^{7}+\frac{185879495}{225524577831}a^{6}-\frac{101105881}{21478531222}a^{5}-\frac{3306456515}{75174859277}a^{4}-\frac{27372473603}{451049155662}a^{3}+\frac{16429975210}{10739265611}a^{2}+\frac{295520651976}{75174859277}a+\frac{3023034124991}{225524577831}$, $\frac{67682089}{150349718554}a^{7}-\frac{40347192}{75174859277}a^{6}-\frac{787702145}{21478531222}a^{5}+\frac{4664133268}{75174859277}a^{4}+\frac{12803138979}{150349718554}a^{3}+\frac{253331999983}{75174859277}a^{2}+\frac{572419280340}{75174859277}a+\frac{964530442601}{75174859277}$ 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:  \( 612.159317198 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 612.159317198 \cdot 80}{2\cdot\sqrt{1046738867982336}}\cr\approx \mathstrut & 1.17957281016 \end{aligned}\]

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^8 - 2*x^7 - 41*x^6 - 218*x^5 + 865*x^4 + 5296*x^3 + 36682*x^2 + 71584*x + 99628) 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 - 2*x^7 - 41*x^6 - 218*x^5 + 865*x^4 + 5296*x^3 + 36682*x^2 + 71584*x + 99628, 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 - 2*x^7 - 41*x^6 - 218*x^5 + 865*x^4 + 5296*x^3 + 36682*x^2 + 71584*x + 99628); 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 - 2*x^7 - 41*x^6 - 218*x^5 + 865*x^4 + 5296*x^3 + 36682*x^2 + 71584*x + 99628); 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

$S_4$ (as 8T14):

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 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

\(\Q(\sqrt{-474}) \), 4.2.17064.1

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

Galois closure: deg 24
Degree 4 sibling: 4.2.17064.1
Degree 6 siblings: 6.0.552077462016.4, 6.2.291180096.5
Degree 12 siblings: deg 12, deg 12
Minimal sibling: 4.2.17064.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{4}$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
2.2.2.6a1.1$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 3$$2$$2$$6$$C_2^2$$$[3]^{2}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.6.7a1.3$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
\(79\) Copy content Toggle raw display 79.2.2.2a1.2$x^{4} + 156 x^{3} + 6090 x^{2} + 468 x + 88$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
79.2.2.2a1.2$x^{4} + 156 x^{3} + 6090 x^{2} + 468 x + 88$$2$$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)