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Properties

Label 8.0.248669678479753216.4
Degree $8$
Signature $[0, 4]$
Discriminant $2.487\times 10^{17}$
Root discriminant \(149.44\)
Ramified primes $2,7,43$
Class number $8$ (GRH)
Class group [2, 2, 2] (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 - 2*x^7 + 58*x^6 + 360*x^5 + 888*x^4 + 428*x^3 + 7960*x^2 - 13984*x + 7226)
 
Copy content gp:K = bnfinit(y^8 - 2*y^7 + 58*y^6 + 360*y^5 + 888*y^4 + 428*y^3 + 7960*y^2 - 13984*y + 7226, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 + 58*x^6 + 360*x^5 + 888*x^4 + 428*x^3 + 7960*x^2 - 13984*x + 7226);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 + 58*x^6 + 360*x^5 + 888*x^4 + 428*x^3 + 7960*x^2 - 13984*x + 7226)
 

\( x^{8} - 2x^{7} + 58x^{6} + 360x^{5} + 888x^{4} + 428x^{3} + 7960x^{2} - 13984x + 7226 \) 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:   \(248669678479753216\) \(\medspace = 2^{14}\cdot 7^{4}\cdot 43^{6}\) 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:  \(149.44\)
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:  $2^{7/4}7^{1/2}43^{6/7}\approx 223.5976997666865$
Ramified primes:   \(2\), \(7\), \(43\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5004550054349}a^{7}+\frac{761276254266}{5004550054349}a^{6}-\frac{2375242835212}{5004550054349}a^{5}+\frac{1260646864656}{5004550054349}a^{4}+\frac{1682220098885}{5004550054349}a^{3}+\frac{1286315688354}{5004550054349}a^{2}-\frac{1838979245220}{5004550054349}a+\frac{560732642010}{5004550054349}$ 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:  $C_{2}\times C_{2}\times C_{2}$, 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:  $C_{2}\times C_{2}\times C_{2}$, 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:   $\frac{223129423852}{5004550054349}a^{7}-\frac{795838475818}{5004550054349}a^{6}+\frac{2529661633618}{5004550054349}a^{5}+\frac{40086691696648}{5004550054349}a^{4}+\frac{141730715548568}{5004550054349}a^{3}+\frac{51696046710146}{5004550054349}a^{2}+\frac{12\cdots 32}{5004550054349}a-\frac{10\cdots 15}{5004550054349}$, $\frac{17\cdots 41}{5004550054349}a^{7}-\frac{11\cdots 29}{5004550054349}a^{6}+\frac{14\cdots 12}{5004550054349}a^{5}+\frac{10\cdots 44}{5004550054349}a^{4}+\frac{12\cdots 02}{5004550054349}a^{3}+\frac{25\cdots 00}{5004550054349}a^{2}-\frac{42\cdots 80}{5004550054349}a+\frac{21\cdots 27}{5004550054349}$, $\frac{10\cdots 64}{5004550054349}a^{7}+\frac{51\cdots 20}{5004550054349}a^{6}+\frac{45\cdots 50}{5004550054349}a^{5}+\frac{33\cdots 50}{5004550054349}a^{4}+\frac{70\cdots 48}{5004550054349}a^{3}-\frac{14\cdots 90}{5004550054349}a^{2}+\frac{64\cdots 36}{5004550054349}a+\frac{10\cdots 25}{5004550054349}$ 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:  \( 87970.1931263 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 87970.1931263 \cdot 8}{2\cdot\sqrt{248669678479753216}}\cr\approx \mathstrut & 1.09977436854 \end{aligned}\] (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 - 2*x^7 + 58*x^6 + 360*x^5 + 888*x^4 + 428*x^3 + 7960*x^2 - 13984*x + 7226) 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 + 58*x^6 + 360*x^5 + 888*x^4 + 428*x^3 + 7960*x^2 - 13984*x + 7226, 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 + 58*x^6 + 360*x^5 + 888*x^4 + 428*x^3 + 7960*x^2 - 13984*x + 7226); 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 - 2*x^7 + 58*x^6 + 360*x^5 + 888*x^4 + 428*x^3 + 7960*x^2 - 13984*x + 7226); 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 R ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ R ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.1.0.1}{1} }^{8}$ ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ R ${\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\) Copy content Toggle raw display 2.1.8.14a2.1$x^{8} + 2 x^{7} + 2 x^{6} + 2$$8$$1$$14$$C_2^3:C_7$$$[2, 2, 2]^{7}$$
\(7\) Copy content Toggle raw display 7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
7.2.2.2a1.2$x^{4} + 12 x^{3} + 42 x^{2} + 36 x + 16$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(43\) Copy content Toggle raw display $\Q_{43}$$x + 40$$1$$1$$0$Trivial$$[\ ]$$
43.1.7.6a1.1$x^{7} + 43$$7$$1$$6$$C_7$$$[\ ]_{7}$$

Spectrum of ring of integers

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