LMFDB - Number field 8.4.2396865997.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67)

gp:K = bnfinit(y^8 - 2*y^7 - 7*y^6 + 24*y^5 - 2*y^4 - 21*y^3 - 21*y^2 - 22*y + 67, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^8 - 2*x^7 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^8 - 2*x^7 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67)

$x^{8} - 2x^{7} - 7x^{6} + 24x^{5} - 2x^{4} - 21x^{3} - 21x^{2} - 22x + 67$ x^8 - 2*x^7 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$8$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[4, 2]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$2396865997$ 2396865997 $\medspace = 131\cdot 167\cdot 331^{2}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$14.87$	sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:(1.0 * dK)^(1/degree(K))
Galois root discriminant:	$131^{1/2}167^{1/2}331^{1/2}\approx 2690.963953679053$
Ramified primes:	$131$ , $167$ , $331$ 131, 167, 331	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{21877})$
$\Aut(K/\Q)$ :	$C_2$	comment:Autmorphisms sage:K.automorphisms() magma:Automorphisms(K); 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 $a$ )

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $\frac{1}{19}a^{6}+\frac{7}{19}a^{5}-\frac{7}{19}a^{4}-\frac{5}{19}a^{3}-\frac{5}{19}a^{2}+\frac{2}{19}a+\frac{8}{19}$ , $\frac{1}{8303}a^{7}-\frac{102}{8303}a^{6}+\frac{1890}{8303}a^{5}+\frac{1993}{8303}a^{4}-\frac{30}{8303}a^{3}+\frac{2979}{8303}a^{2}+\frac{987}{8303}a+\frac{914}{8303}$ 1, a, a^2, a^3, a^4, a^5, 1/19*a^6 + 7/19*a^5 - 7/19*a^4 - 5/19*a^3 - 5/19*a^2 + 2/19*a + 8/19, 1/8303*a^7 - 102/8303*a^6 + 1890/8303*a^5 + 1993/8303*a^4 - 30/8303*a^3 + 2979/8303*a^2 + 987/8303*a + 914/8303

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		Trivial group, which has order $1$	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$5$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$-1$ -1 (order $2$ )	comment:Generator for roots of unity sage:UK.torsion_generator() gp:K.tu[2] magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar:torsion_units_generator(OK)
Fundamental units:	$\frac{876}{8303}a^{7}-\frac{641}{8303}a^{6}-\frac{6708}{8303}a^{5}+\frac{12289}{8303}a^{4}+\frac{11739}{8303}a^{3}-\frac{1034}{8303}a^{2}-\frac{12447}{8303}a-\frac{34006}{8303}$ , $\frac{1366}{8303}a^{7}-\frac{803}{8303}a^{6}-\frac{10544}{8303}a^{5}+\frac{17408}{8303}a^{4}+\frac{21948}{8303}a^{3}+\frac{5651}{8303}a^{2}-\frac{26997}{8303}a-\frac{67720}{8303}$ , $\frac{614}{8303}a^{7}-\frac{137}{8303}a^{6}-\frac{4582}{8303}a^{5}+\frac{5783}{8303}a^{4}+\frac{9548}{8303}a^{3}+\frac{13808}{8303}a^{2}-\frac{7967}{8303}a-\frac{51478}{8303}$ , $\frac{1366}{8303}a^{7}-\frac{803}{8303}a^{6}-\frac{10544}{8303}a^{5}+\frac{17408}{8303}a^{4}+\frac{21948}{8303}a^{3}+\frac{5651}{8303}a^{2}-\frac{18694}{8303}a-\frac{59417}{8303}$ , $\frac{33}{8303}a^{7}+\frac{567}{8303}a^{6}-\frac{1432}{8303}a^{5}-\frac{3277}{8303}a^{4}+\frac{12557}{8303}a^{3}-\frac{4388}{8303}a^{2}+\frac{7225}{8303}a-\frac{21404}{8303}$ 876/8303a^7 - 641/8303a^6 - 6708/8303a^5 + 12289/8303a^4 + 11739/8303a^3 - 1034/8303a^2 - 12447/8303a - 34006/8303, 1366/8303a^7 - 803/8303a^6 - 10544/8303a^5 + 17408/8303a^4 + 21948/8303a^3 + 5651/8303a^2 - 26997/8303a - 67720/8303, 614/8303a^7 - 137/8303a^6 - 4582/8303a^5 + 5783/8303a^4 + 9548/8303a^3 + 13808/8303a^2 - 7967/8303a - 51478/8303, 1366/8303a^7 - 803/8303a^6 - 10544/8303a^5 + 17408/8303a^4 + 21948/8303a^3 + 5651/8303a^2 - 18694/8303a - 59417/8303, 33/8303a^7 + 567/8303a^6 - 1432/8303a^5 - 3277/8303a^4 + 12557/8303a^3 - 4388/8303a^2 + 7225/8303*a - 21404/8303	comment:Fundamental units sage:UK.fundamental_units() gp:K.fu magma:[K\|fUK(g): g in Generators(UK)]; oscar:[K(fUK(a)) for a in gens(UK)]
Regulator:	$31.916130481$	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 31.916130481 \cdot 1}{2\cdot\sqrt{2396865997}}\cr\approx \mathstrut & 0.20589133920 \end{aligned}$

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^8 - 2*x^7 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^8 - 2*x^7 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67, 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)))]

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 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67); 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)));

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 - 7*x^6 + 24*x^5 - 2*x^4 - 21*x^3 - 21*x^2 - 22*x + 67); 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

$C_2\wr S_4$ (as 8T44):

comment:Galois group

sage:K.galois_group(type='pari')

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 384

The 20 conjugacy class representatives for

C_2 \wr S_4

Character table for

C_2 \wr S_4

Intermediate fields

4.2.331.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(b)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 8 siblings:	data not computed
Degree 16 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 32 siblings:	data not computed
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	${\href{/padicField/2.8.0.1}{8} }$	${\href{/padicField/3.4.0.1}{4} }^{2}$	${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$	${\href{/padicField/7.4.0.1}{4} }^{2}$	${\href{/padicField/11.4.0.1}{4} }^{2}$	${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$	${\href{/padicField/23.2.0.1}{2} }^{4}$	${\href{/padicField/29.8.0.1}{8} }$	${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$	${\href{/padicField/37.4.0.1}{4} }^{2}$	${\href{/padicField/41.8.0.1}{8} }$	${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

comment:Frobenius cycle types

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)]

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]])

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))];

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$131$ 131	131.1.2.1a1.1	$x^{2} + 131$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$131$ 131	131.6.1.0a1.1	$x^{6} + 2 x^{4} + 66 x^{3} + 4 x^{2} + 22 x + 2$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$167$ 167	167.1.2.1a1.2	$x^{2} + 835$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$167$ 167	167.6.1.0a1.1	$x^{6} + 2 x^{4} + 75 x^{3} + 38 x^{2} + 2 x + 5$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$331$ 331		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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Normalized defining polynomial

Invariants

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

Class group and class number

Unit group

Class number formula

Galois group

Intermediate fields

Sibling fields

Frobenius cycle types

Local algebras for ramified primes

Spectrum of ring of integers

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	8.4.2396865997.1

Degree	$8$
Signature	$[4, 2]$
Discriminant	$2396865997$
Root discriminant	$14.87$
Ramified primes	$131,167,331$
Class number	$1$
Class group	trivial
Galois group	$C_2 \wr S_4$ (as 8T44)

Properties

Related objects

Downloads

Learn more

Integral basis (with respect to field generator aaa)

Class group and class number

Sibling fields

Local algebras for ramified primes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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