LMFDB - Number field 10.0.1538775332.1

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

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*x + 1)

gp:K = bnfinit(y^10 - 3*y^9 + 3*y^8 - 2*y^7 + 5*y^6 - 6*y^5 + 5*y^4 - 2*y^3 + 3*y^2 - 3*y + 1, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*x + 1);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*x + 1)

$x^{10} - 3x^{9} + 3x^{8} - 2x^{7} + 5x^{6} - 6x^{5} + 5x^{4} - 2x^{3} + 3x^{2} - 3x + 1$ x^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*x + 1

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$10$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$[0, 5]$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$-1538775332$ -1538775332 $\medspace = -\,2^{2}\cdot 17\cdot 67^{2}\cdot 71^{2}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$8.29$	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:	$2\cdot 17^{1/2}67^{1/2}71^{1/2}\approx 568.7495054942905$
Ramified primes:	$2$ , $17$ , $67$ , $71$ 2, 17, 67, 71	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:	$\Q(\sqrt{-17})$
$\Aut(K/\Q)$ :	$C_2$	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 $a$ )

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $a^{6}$ , $a^{7}$ , $a^{8}$ , $a^{9}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		Yes
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:	$4$	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:	$2a^{9}-5a^{8}+2a^{7}+3a^{6}+2a^{5}-3a^{4}+a^{3}+5a^{2}-a-1$ , $a$ , $2a^{9}-6a^{8}+6a^{7}-4a^{6}+9a^{5}-9a^{4}+7a^{3}-3a^{2}+4a-3$ , $a^{8}-3a^{7}+3a^{6}-a^{5}+2a^{4}-3a^{3}+3a^{2}+a$ 2a^9 - 5a^8 + 2a^7 + 3a^6 + 2a^5 - 3a^4 + a^3 + 5a^2 - a - 1, a, 2a^9 - 6a^8 + 6a^7 - 4a^6 + 9a^5 - 9a^4 + 7a^3 - 3a^2 + 4a - 3, a^8 - 3a^7 + 3a^6 - a^5 + 2a^4 - 3a^3 + 3*a^2 + a	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:	$3.11760635261$	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 3.11760635261 \cdot 1}{2\cdot\sqrt{1538775332}}\cr\approx \mathstrut & 0.389137438730 \end{aligned}$

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^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 3*x^8 - 2*x^7 + 5*x^6 - 6*x^5 + 5*x^4 - 2*x^3 + 3*x^2 - 3*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

$C_2\wr S_5$ (as 10T39):

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 non-solvable group of order 3840

The 36 conjugacy class representatives for

C_2 \wr S_5

Character table for

C_2 \wr S_5

Intermediate fields

5.1.4757.1

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 10 sibling:	data not computed
Degree 20 siblings:	data not computed
Degree 30 siblings:	data not computed
Degree 32 siblings:	data not computed
Degree 40 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	R	${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }$	${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$	${\href{/padicField/7.5.0.1}{5} }^{2}$	${\href{/padicField/11.5.0.1}{5} }^{2}$	${\href{/padicField/13.5.0.1}{5} }^{2}$	R	${\href{/padicField/19.10.0.1}{10} }$	${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.10.0.1}{10} }$	${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$	${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. 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
$2$ 2	2.1.2.2a1.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.4.1.0a1.1	$x^{4} + x + 1$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	2.4.1.0a1.1	$x^{4} + x + 1$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$17$ 17	17.1.2.1a1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$17$ 17	17.8.1.0a1.1	$x^{8} + 11 x^{4} + 12 x^{3} + 6 x + 3$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$
$67$ 67	$\Q_{67}$	$x + 65$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{67}$	$x + 65$	$1$	$1$	$0$	Trivial	$[\ ]$
	67.2.2.2a1.2	$x^{4} + 126 x^{3} + 3973 x^{2} + 252 x + 71$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	67.4.1.0a1.1	$x^{4} + 8 x^{2} + 54 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$71$ 71	71.1.2.1a1.1	$x^{2} + 71$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.1.2.1a1.1	$x^{2} + 71$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	71.3.1.0a1.1	$x^{3} + 4 x + 64$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	71.3.1.0a1.1	$x^{3} + 4 x + 64$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$

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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	10.0.1538775332.1

Degree	$10$
Signature	$[0, 5]$
Discriminant	$-1538775332$
Root discriminant	$8.29$
Ramified primes	$2,17,67,71$
Class number	$1$
Class group	trivial
Galois group	$C_2 \wr S_5$ (as 10T39)

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 $a$ )