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Properties

Label 8.8.116507435287321.1
Degree $8$
Signature $(8, 0)$
Discriminant $1.165\times 10^{14}$
Root discriminant \(57.32\)
Ramified primes $13,17$
Class number $2$
Class group [2]
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 - 67*x^6 - 16*x^5 + 863*x^4 + 1276*x^3 + 392*x^2 - 54*x + 1)
 
Copy content gp:K = bnfinit(y^8 - 3*y^7 - 67*y^6 - 16*y^5 + 863*y^4 + 1276*y^3 + 392*y^2 - 54*y + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 3*x^7 - 67*x^6 - 16*x^5 + 863*x^4 + 1276*x^3 + 392*x^2 - 54*x + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 3*x^7 - 67*x^6 - 16*x^5 + 863*x^4 + 1276*x^3 + 392*x^2 - 54*x + 1)
 

\( x^{8} - 3x^{7} - 67x^{6} - 16x^{5} + 863x^{4} + 1276x^{3} + 392x^{2} - 54x + 1 \) 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:   \(116507435287321\) \(\medspace = 13^{6}\cdot 17^{6}\) 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:  \(57.32\)
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:  $13^{3/4}17^{3/4}\approx 57.318419318377835$
Ramified primes:   \(13\), \(17\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{8232013}a^{7}+\frac{3019361}{8232013}a^{6}-\frac{1122526}{8232013}a^{5}+\frac{2726932}{8232013}a^{4}-\frac{1466398}{8232013}a^{3}-\frac{1137546}{8232013}a^{2}+\frac{2039677}{8232013}a+\frac{1971827}{8232013}$ 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:  $C_{2}$, which has order $2$
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:   $\frac{218396}{8232013}a^{7}-\frac{804396}{8232013}a^{6}-\frac{14073169}{8232013}a^{5}+\frac{6060587}{8232013}a^{4}+\frac{183880430}{8232013}a^{3}+\frac{154832358}{8232013}a^{2}-\frac{18085403}{8232013}a-\frac{2166577}{8232013}$, $\frac{885973}{8232013}a^{7}-\frac{2621227}{8232013}a^{6}-\frac{59397333}{8232013}a^{5}-\frac{17138521}{8232013}a^{4}+\frac{761065628}{8232013}a^{3}+\frac{1171023165}{8232013}a^{2}+\frac{427089624}{8232013}a-\frac{12316189}{8232013}$, $\frac{43328}{8232013}a^{7}-\frac{277188}{8232013}a^{6}-\frac{2073724}{8232013}a^{5}+\frac{6659120}{8232013}a^{4}+\frac{23279829}{8232013}a^{3}-\frac{27227296}{8232013}a^{2}-\frac{11998525}{8232013}a+\frac{11721355}{8232013}$, $\frac{2711054}{8232013}a^{7}-\frac{7818303}{8232013}a^{6}-\frac{182676824}{8232013}a^{5}-\frac{64008582}{8232013}a^{4}+\frac{2339766290}{8232013}a^{3}+\frac{3723034569}{8232013}a^{2}+\frac{1400303304}{8232013}a-\frac{32750373}{8232013}$, $\frac{885973}{8232013}a^{7}-\frac{2621227}{8232013}a^{6}-\frac{59397333}{8232013}a^{5}-\frac{17138521}{8232013}a^{4}+\frac{761065628}{8232013}a^{3}+\frac{1171023165}{8232013}a^{2}+\frac{418857611}{8232013}a-\frac{20548202}{8232013}$, $\frac{149208}{8232013}a^{7}-\frac{559363}{8232013}a^{6}-\frac{9554923}{8232013}a^{5}+\frac{4595318}{8232013}a^{4}+\frac{123840938}{8232013}a^{3}+\frac{103696635}{8232013}a^{2}+\frac{15069232}{8232013}a+\frac{218396}{8232013}$, $\frac{1806}{6731}a^{7}-\frac{5602}{6731}a^{6}-\frac{120148}{6731}a^{5}-\frac{18116}{6731}a^{4}+\frac{1545292}{6731}a^{3}+\frac{2168202}{6731}a^{2}+\frac{668854}{6731}a-\frac{16953}{6731}$ 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:  \( 7915.1159478 \)
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:  \( 6 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 7915.1159478 \cdot 2}{2\cdot\sqrt{116507435287321}}\cr\approx \mathstrut & 0.18772427055 \end{aligned}\]

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 - 67*x^6 - 16*x^5 + 863*x^4 + 1276*x^3 + 392*x^2 - 54*x + 1) 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 - 67*x^6 - 16*x^5 + 863*x^4 + 1276*x^3 + 392*x^2 - 54*x + 1, 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 - 67*x^6 - 16*x^5 + 863*x^4 + 1276*x^3 + 392*x^2 - 54*x + 1); 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 - 67*x^6 - 16*x^5 + 863*x^4 + 1276*x^3 + 392*x^2 - 54*x + 1); 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

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{17})\)

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 ${\href{/padicField/2.4.0.1}{4} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{2}$ ${\href{/padicField/5.4.0.1}{4} }^{2}$ ${\href{/padicField/7.4.0.1}{4} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}$ R R ${\href{/padicField/19.4.0.1}{4} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{2}$ ${\href{/padicField/29.4.0.1}{4} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{2}$ ${\href{/padicField/53.1.0.1}{1} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{2}$

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
\(13\) Copy content Toggle raw display 13.1.4.3a1.3$x^{4} + 52$$4$$1$$3$$C_4$$$[\ ]_{4}$$
13.1.4.3a1.3$x^{4} + 52$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(17\) Copy content Toggle raw display 17.1.4.3a1.3$x^{4} + 153$$4$$1$$3$$C_4$$$[\ ]_{4}$$
17.1.4.3a1.3$x^{4} + 153$$4$$1$$3$$C_4$$$[\ ]_{4}$$

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 1.221.2t1.a.a$1$ $ 13 \cdot 17 $ \(\Q(\sqrt{221}) \) $C_2$ (as 2T1) $1$ $1$
*8 1.17.2t1.a.a$1$ $ 17 $ \(\Q(\sqrt{17}) \) $C_2$ (as 2T1) $1$ $1$
*8 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
*16 2.48841.8t5.a.a$2$ $ 13^{2} \cdot 17^{2}$ 8.8.116507435287321.1 $Q_8$ (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)