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Properties

Label 8.8.151939915084881.1
Degree $8$
Signature $(8, 0)$
Discriminant $1.519\times 10^{14}$
Root discriminant \(59.25\)
Ramified primes $3,7,11$
Class number $1$
Class group trivial
Galois group $Q_8$ (as 8T5)

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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 - 62*x^6 + 66*x^5 + 1125*x^4 + 264*x^3 - 4982*x^2 - 4245*x + 823)
 
Copy content gp:K = bnfinit(y^8 - 3*y^7 - 62*y^6 + 66*y^5 + 1125*y^4 + 264*y^3 - 4982*y^2 - 4245*y + 823, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 3*x^7 - 62*x^6 + 66*x^5 + 1125*x^4 + 264*x^3 - 4982*x^2 - 4245*x + 823);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 3*x^7 - 62*x^6 + 66*x^5 + 1125*x^4 + 264*x^3 - 4982*x^2 - 4245*x + 823)
 

\( x^{8} - 3x^{7} - 62x^{6} + 66x^{5} + 1125x^{4} + 264x^{3} - 4982x^{2} - 4245x + 823 \) 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:  $(8, 0)$
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:   \(151939915084881\) \(\medspace = 3^{6}\cdot 7^{6}\cdot 11^{6}\) 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:  \(59.25\)
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:  $3^{3/4}7^{3/4}11^{3/4}\approx 59.252814612259755$
Ramified primes:   \(3\), \(7\), \(11\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $Q_8$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2353099898}a^{7}+\frac{407547641}{2353099898}a^{6}+\frac{14549161}{35120894}a^{5}-\frac{521753154}{1176549949}a^{4}-\frac{402984125}{1176549949}a^{3}-\frac{698086745}{2353099898}a^{2}-\frac{219533267}{2353099898}a-\frac{199951589}{2353099898}$ 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 $1$
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:  $C_{2}\times C_{2}\times C_{2}$, which has order $8$
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:   $\frac{2771541}{1176549949}a^{7}-\frac{16553179}{1176549949}a^{6}-\frac{1647377}{17560447}a^{5}+\frac{445604385}{1176549949}a^{4}+\frac{1329149119}{1176549949}a^{3}-\frac{2001340842}{1176549949}a^{2}-\frac{3233178638}{1176549949}a+\frac{2176949433}{1176549949}$, $\frac{2693101}{1176549949}a^{7}-\frac{14848940}{1176549949}a^{6}-\frac{1946134}{17560447}a^{5}+\frac{500451975}{1176549949}a^{4}+\frac{1920269502}{1176549949}a^{3}-\frac{4241339349}{1176549949}a^{2}-\frac{6558418569}{1176549949}a+\frac{8851520168}{1176549949}$, $\frac{3959654}{1176549949}a^{7}-\frac{27322543}{1176549949}a^{6}-\frac{1779362}{17560447}a^{5}+\frac{662845719}{1176549949}a^{4}+\frac{1035178530}{1176549949}a^{3}-\frac{2581305324}{1176549949}a^{2}-\frac{3026890050}{1176549949}a+\frac{177640211}{1176549949}$, $\frac{13503205}{2353099898}a^{7}-\frac{93962991}{2353099898}a^{6}-\frac{6923015}{35120894}a^{5}+\frac{1307546494}{1176549949}a^{4}+\frac{2735552794}{1176549949}a^{3}-\frac{14948424319}{2353099898}a^{2}-\frac{21578417989}{2353099898}a+\frac{4235520517}{2353099898}$, $\frac{10722247}{2353099898}a^{7}-\frac{47611471}{2353099898}a^{6}-\frac{8117281}{35120894}a^{5}+\frac{573333521}{1176549949}a^{4}+\frac{4363722758}{1176549949}a^{3}-\frac{683271385}{2353099898}a^{2}-\frac{37023523589}{2353099898}a-\frac{32773732173}{2353099898}$, $\frac{1304995}{1176549949}a^{7}-\frac{5179165}{1176549949}a^{6}-\frac{1223016}{17560447}a^{5}+\frac{163412763}{1176549949}a^{4}+\frac{1702899392}{1176549949}a^{3}-\frac{615930422}{1176549949}a^{2}-\frac{11668686655}{1176549949}a-\frac{11165147376}{1176549949}$, $\frac{4552052}{1176549949}a^{7}-\frac{22524123}{1176549949}a^{6}-\frac{3574695}{17560447}a^{5}+\frac{721703225}{1176549949}a^{4}+\frac{3967168669}{1176549949}a^{3}-\frac{4584395722}{1176549949}a^{2}-\frac{15912081091}{1176549949}a+\frac{1951985211}{1176549949}$ 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:  \( 17804.5169355 \)
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:  \( 5 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 17804.5169355 \cdot 1}{2\cdot\sqrt{151939915084881}}\cr\approx \mathstrut & 0.184886084479 \end{aligned}\]

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 - 3*x^7 - 62*x^6 + 66*x^5 + 1125*x^4 + 264*x^3 - 4982*x^2 - 4245*x + 823) 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 - 62*x^6 + 66*x^5 + 1125*x^4 + 264*x^3 - 4982*x^2 - 4245*x + 823, 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 - 62*x^6 + 66*x^5 + 1125*x^4 + 264*x^3 - 4982*x^2 - 4245*x + 823); 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 - 62*x^6 + 66*x^5 + 1125*x^4 + 264*x^3 - 4982*x^2 - 4245*x + 823); 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

$Q_8$ (as 8T5):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
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 8
The 5 conjugacy class representatives for $Q_8$
Character table for $Q_8$

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{21}, \sqrt{33})\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{2}$ R ${\href{/padicField/5.4.0.1}{4} }^{2}$ R R ${\href{/padicField/13.4.0.1}{4} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(3\) Copy content Toggle raw display 3.2.4.6a1.3$x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 19$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(7\) Copy content Toggle raw display 7.2.4.6a1.3$x^{8} + 24 x^{7} + 228 x^{6} + 1080 x^{5} + 2646 x^{4} + 3240 x^{3} + 2052 x^{2} + 655 x + 109$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$
\(11\) Copy content Toggle raw display 11.2.4.6a1.3$x^{8} + 28 x^{7} + 302 x^{6} + 1540 x^{5} + 3601 x^{4} + 3080 x^{3} + 1208 x^{2} + 268 x + 115$$4$$2$$6$$Q_8$$$[\ ]_{4}^{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*8 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
*8 1.33.2t1.a.a$1$ $ 3 \cdot 11 $ \(\Q(\sqrt{33}) \) $C_2$ (as 2T1) $1$ $1$
*8 1.21.2t1.a.a$1$ $ 3 \cdot 7 $ \(\Q(\sqrt{21}) \) $C_2$ (as 2T1) $1$ $1$
*8 1.77.2t1.a.a$1$ $ 7 \cdot 11 $ \(\Q(\sqrt{77}) \) $C_2$ (as 2T1) $1$ $1$
*16 2.53361.8t5.a.a$2$ $ 3^{2} \cdot 7^{2} \cdot 11^{2}$ 8.8.151939915084881.1 $Q_8$ (as 8T5) $-1$ $2$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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