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Properties

Label 8.0.10699063813365841.1
Degree $8$
Signature $[0, 4]$
Discriminant $1.070\times 10^{16}$
Root discriminant \(100.85\)
Ramified primes $17,71$
Class number $8$ (GRH)
Class group [2, 4] (GRH)
Galois group $C_2^3:C_7$ (as 8T25)

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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 - x^7 + 41*x^6 - 98*x^5 + 411*x^4 - 2087*x^3 + 7263*x^2 - 8915*x + 6934)
 
Copy content gp:K = bnfinit(y^8 - y^7 + 41*y^6 - 98*y^5 + 411*y^4 - 2087*y^3 + 7263*y^2 - 8915*y + 6934, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - x^7 + 41*x^6 - 98*x^5 + 411*x^4 - 2087*x^3 + 7263*x^2 - 8915*x + 6934);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - x^7 + 41*x^6 - 98*x^5 + 411*x^4 - 2087*x^3 + 7263*x^2 - 8915*x + 6934)
 

\( x^{8} - x^{7} + 41x^{6} - 98x^{5} + 411x^{4} - 2087x^{3} + 7263x^{2} - 8915x + 6934 \) 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:   \(10699063813365841\) \(\medspace = 17^{4}\cdot 71^{6}\) 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:  \(100.85\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $17^{1/2}71^{6/7}\approx 159.2272267235735$
Ramified primes:   \(17\), \(71\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17}a^{6}-\frac{5}{17}a^{5}+\frac{6}{17}a^{4}-\frac{4}{17}a^{2}+\frac{3}{17}a+\frac{8}{17}$, $\frac{1}{5745631831}a^{7}+\frac{12562096}{5745631831}a^{6}+\frac{800345223}{5745631831}a^{5}+\frac{186636603}{5745631831}a^{4}-\frac{2571143924}{5745631831}a^{3}+\frac{796710996}{5745631831}a^{2}-\frac{224812644}{5745631831}a-\frac{1624706253}{5745631831}$ 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:  $C_{2}\times C_{4}$, which has order $8$ (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:  $C_{2}\times C_{4}$, which has order $8$ (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:   \( -1 \)  (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:   $\frac{2684806373}{5745631831}a^{7}-\frac{7344527254}{5745631831}a^{6}+\frac{103972111017}{5745631831}a^{5}-\frac{451576880457}{5745631831}a^{4}+\frac{1131941423598}{5745631831}a^{3}-\frac{6784701283181}{5745631831}a^{2}+\frac{25802984068608}{5745631831}a-\frac{38391724281253}{5745631831}$, $\frac{3263221}{337978343}a^{7}+\frac{39037879}{337978343}a^{6}+\frac{47732558}{337978343}a^{5}+\frac{363078660}{337978343}a^{4}-\frac{1227260476}{337978343}a^{3}+\frac{5352430853}{337978343}a^{2}-\frac{11486286903}{337978343}a+\frac{41596754957}{337978343}$, $\frac{8088434519}{5745631831}a^{7}+\frac{8775584323}{5745631831}a^{6}+\frac{335888645586}{5745631831}a^{5}-\frac{156166929322}{5745631831}a^{4}+\frac{2254295785441}{5745631831}a^{3}-\frac{14265160575307}{5745631831}a^{2}+\frac{20779223313028}{5745631831}a-\frac{26343032003141}{5745631831}$ 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:  \( 15761.2152227 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 15761.2152227 \cdot 8}{2\cdot\sqrt{10699063813365841}}\cr\approx \mathstrut & 0.949940218749 \end{aligned}\] (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 - x^7 + 41*x^6 - 98*x^5 + 411*x^4 - 2087*x^3 + 7263*x^2 - 8915*x + 6934) 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 - x^7 + 41*x^6 - 98*x^5 + 411*x^4 - 2087*x^3 + 7263*x^2 - 8915*x + 6934, 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 - x^7 + 41*x^6 - 98*x^5 + 411*x^4 - 2087*x^3 + 7263*x^2 - 8915*x + 6934); 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 = PolynomialRing(QQ); K, a = NumberField(x^8 - x^7 + 41*x^6 - 98*x^5 + 411*x^4 - 2087*x^3 + 7263*x^2 - 8915*x + 6934); 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

$F_8$ (as 8T25):

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); G, transitive_group_identification(G)
 
A solvable group of order 56
The 8 conjugacy class representatives for $C_2^3:C_7$
Character table for $C_2^3:C_7$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 14 sibling: deg 14
Degree 28 sibling: deg 28
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 ${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.2.0.1}{2} }^{4}$ ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ R ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(17\) Copy content Toggle raw display 17.2.2.2a1.2$x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
17.2.2.2a1.2$x^{4} + 32 x^{3} + 262 x^{2} + 96 x + 26$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(71\) Copy content Toggle raw display $\Q_{71}$$x + 64$$1$$1$$0$Trivial$$[\ ]$$
71.1.7.6a1.1$x^{7} + 71$$7$$1$$6$$C_7$$$[\ ]_{7}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.71.7t1.a.a$1$ $ 71 $ 7.7.128100283921.1 $C_7$ (as 7T1) $0$ $1$
1.71.7t1.a.b$1$ $ 71 $ 7.7.128100283921.1 $C_7$ (as 7T1) $0$ $1$
1.71.7t1.a.c$1$ $ 71 $ 7.7.128100283921.1 $C_7$ (as 7T1) $0$ $1$
1.71.7t1.a.d$1$ $ 71 $ 7.7.128100283921.1 $C_7$ (as 7T1) $0$ $1$
1.71.7t1.a.e$1$ $ 71 $ 7.7.128100283921.1 $C_7$ (as 7T1) $0$ $1$
1.71.7t1.a.f$1$ $ 71 $ 7.7.128100283921.1 $C_7$ (as 7T1) $0$ $1$
* 7.106...841.8t25.a.a$7$ $ 17^{4} \cdot 71^{6}$ 8.0.10699063813365841.1 $C_2^3:C_7$ (as 8T25) $1$ $-1$

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)