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Properties

Label 8.4.3475049209404304.1
Degree $8$
Signature $[4, 2]$
Discriminant $3.475\times 10^{15}$
Root discriminant \(87.62\)
Ramified primes $2,7,67$
Class number $3$ (GRH)
Class group [3] (GRH)
Galois group $S_4\times C_2$ (as 8T24)

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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 - 4*x^7 - 24*x^6 + 86*x^5 + 215*x^4 - 578*x^3 - 1350*x^2 + 1654*x + 1443)
 
Copy content gp:K = bnfinit(y^8 - 4*y^7 - 24*y^6 + 86*y^5 + 215*y^4 - 578*y^3 - 1350*y^2 + 1654*y + 1443, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 4*x^7 - 24*x^6 + 86*x^5 + 215*x^4 - 578*x^3 - 1350*x^2 + 1654*x + 1443);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 4*x^7 - 24*x^6 + 86*x^5 + 215*x^4 - 578*x^3 - 1350*x^2 + 1654*x + 1443)
 

\( x^{8} - 4x^{7} - 24x^{6} + 86x^{5} + 215x^{4} - 578x^{3} - 1350x^{2} + 1654x + 1443 \) 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:  $[4, 2]$
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:   \(3475049209404304\) \(\medspace = 2^{4}\cdot 7^{4}\cdot 67^{6}\) 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:  \(87.62\)
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:  $2^{2/3}7^{1/2}67^{3/4}\approx 98.35394535093855$
Ramified primes:   \(2\), \(7\), \(67\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
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.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{196}a^{6}-\frac{3}{196}a^{5}+\frac{12}{49}a^{4}-\frac{13}{28}a^{3}+\frac{45}{196}a+\frac{67}{196}$, $\frac{1}{135828}a^{7}+\frac{1}{396}a^{6}-\frac{5297}{67914}a^{5}-\frac{3043}{15092}a^{4}+\frac{2329}{9702}a^{3}+\frac{15725}{135828}a^{2}+\frac{22105}{135828}a-\frac{10183}{22638}$ 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:  $2$, $3$

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$ (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:  $C_{6}$, which has order $6$ (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:  $5$
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:   $\frac{620}{33957}a^{7}-\frac{211}{4851}a^{6}-\frac{17351}{33957}a^{5}+\frac{5763}{7546}a^{4}+\frac{25177}{4851}a^{3}-\frac{162103}{67914}a^{2}-\frac{990658}{33957}a-\frac{374909}{22638}$, $\frac{24}{3773}a^{7}-\frac{5}{154}a^{6}-\frac{926}{3773}a^{5}+\frac{1909}{3773}a^{4}+\frac{1836}{539}a^{3}+\frac{7646}{3773}a^{2}-\frac{32196}{3773}a-\frac{37735}{7546}$, $\frac{271}{67914}a^{7}-\frac{185}{9702}a^{6}-\frac{3749}{33957}a^{5}+\frac{2280}{3773}a^{4}+\frac{1915}{4851}a^{3}-\frac{161360}{33957}a^{2}+\frac{187273}{67914}a+\frac{84181}{22638}$, $\frac{569}{67914}a^{7}+\frac{31}{4851}a^{6}-\frac{10667}{67914}a^{5}-\frac{661}{7546}a^{4}+\frac{10765}{9702}a^{3}+\frac{118705}{67914}a^{2}-\frac{62126}{33957}a-\frac{45415}{22638}$, $\frac{40\cdots 73}{67914}a^{7}-\frac{349255803558982}{693}a^{6}-\frac{14\cdots 43}{67914}a^{5}+\frac{50\cdots 54}{3773}a^{4}+\frac{23\cdots 93}{9702}a^{3}-\frac{24\cdots 40}{33957}a^{2}-\frac{17\cdots 31}{33957}a+\frac{33\cdots 85}{11319}$ 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:  \( 226076.711964 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 226076.711964 \cdot 3}{2\cdot\sqrt{3475049209404304}}\cr\approx \mathstrut & 3.63367706047 \end{aligned}\] (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 - 4*x^7 - 24*x^6 + 86*x^5 + 215*x^4 - 578*x^3 - 1350*x^2 + 1654*x + 1443) 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 - 4*x^7 - 24*x^6 + 86*x^5 + 215*x^4 - 578*x^3 - 1350*x^2 + 1654*x + 1443, 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 - 4*x^7 - 24*x^6 + 86*x^5 + 215*x^4 - 578*x^3 - 1350*x^2 + 1654*x + 1443); 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 - 4*x^7 - 24*x^6 + 86*x^5 + 215*x^4 - 578*x^3 - 1350*x^2 + 1654*x + 1443); 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

$C_2\times S_4$ (as 8T24):

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 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

\(\Q(\sqrt{469}) \), 4.2.58949548.1

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 6 siblings: 6.2.33685456.1, 6.0.2256925552.1
Degree 8 sibling: 8.0.3475049209404304.1
Degree 12 siblings: deg 12, deg 12, deg 12, deg 12, deg 12, deg 12
Degree 16 sibling: deg 16
Degree 24 siblings: deg 24, deg 24, deg 24, deg 24
Minimal sibling: 6.2.33685456.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(7\) Copy content Toggle raw display 7.1.2.1a1.2$x^{2} + 21$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.1.2.1a1.2$x^{2} + 21$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(67\) Copy content Toggle raw display 67.2.4.6a1.2$x^{8} + 252 x^{7} + 23822 x^{6} + 1001700 x^{5} + 15848241 x^{4} + 2003400 x^{3} + 95288 x^{2} + 2016 x + 83$$4$$2$$6$$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)