LMFDB - Number field 12.6.158072000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 1)

gp: K = bnfinit(y^12 - 4*y^9 - 5*y^8 + 6*y^6 + 10*y^5 + 5*y^4 - 4*y^3 - 5*y^2 + 1, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 1);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 1)

$x^{12} - 4x^{9} - 5x^{8} + 6x^{6} + 10x^{5} + 5x^{4} - 4x^{3} - 5x^{2} + 1$ x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 1

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$12$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[6, 3]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-158072000000000$ -158072000000000 $\medspace = -\,2^{12}\cdot 5^{9}\cdot 19759$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$15.25$	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:	$2\cdot 5^{3/4}19759^{1/2}\approx 940.0262460954837$
Ramified primes:	$2$ , $5$ , $19759$ 2, 5, 19759	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-98795})$
$\Aut(K/\Q)$ :	$C_1$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$ .
This is not a CM field.

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

$1$ , $a$ , $a^{2}$ , $a^{3}$ , $a^{4}$ , $a^{5}$ , $a^{6}$ , $a^{7}$ , $a^{8}$ , $a^{9}$ , $a^{10}$ , $a^{11}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11

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

Trivial group, which has order $1$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$8$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$-1$ -1 (order $2$ )	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:	$a$ , $a^{10}-4a^{7}-5a^{6}+5a^{4}+10a^{3}+5a^{2}-2a-2$ , $a^{11}-a^{10}+a^{9}-5a^{8}+6a^{5}+5a^{4}-4a^{2}-3a+1$ , $2a^{11}-a^{10}+a^{9}-9a^{8}-5a^{7}+11a^{5}+15a^{4}+5a^{3}-6a^{2}-5a+1$ , $a^{10}-a^{9}+a^{8}-5a^{7}+5a^{4}+5a^{3}-2a$ , $3a^{11}-a^{10}+a^{9}-12a^{8}-11a^{7}+a^{6}+13a^{5}+24a^{4}+11a^{3}-8a^{2}-6a+1$ , $2a^{11}-a^{10}+a^{9}-9a^{8}-5a^{7}+11a^{5}+15a^{4}+5a^{3}-6a^{2}-5a$ , $a^{10}-4a^{7}-5a^{6}-a^{5}+6a^{4}+9a^{3}+7a^{2}-2a-4$ a, a^10 - 4a^7 - 5a^6 + 5a^4 + 10a^3 + 5a^2 - 2a - 2, a^11 - a^10 + a^9 - 5a^8 + 6a^5 + 5a^4 - 4a^2 - 3a + 1, 2a^11 - a^10 + a^9 - 9a^8 - 5a^7 + 11a^5 + 15a^4 + 5a^3 - 6a^2 - 5a + 1, a^10 - a^9 + a^8 - 5a^7 + 5a^4 + 5a^3 - 2a, 3a^11 - a^10 + a^9 - 12a^8 - 11a^7 + a^6 + 13a^5 + 24a^4 + 11a^3 - 8a^2 - 6a + 1, 2a^11 - a^10 + a^9 - 9a^8 - 5a^7 + 11a^5 + 15a^4 + 5a^3 - 6a^2 - 5a, a^10 - 4a^7 - 5a^6 - a^5 + 6a^4 + 9a^3 + 7a^2 - 2*a - 4	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:	$356.187361644$	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{3}\cdot 356.187361644 \cdot 1}{2\cdot\sqrt{158072000000000}}\cr\approx \mathstrut & 0.224874709286 \end{aligned}$

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 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))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 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)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 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)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^9 - 5*x^8 + 6*x^6 + 10*x^5 + 5*x^4 - 4*x^3 - 5*x^2 + 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

$S_3\wr C_4$ (as 12T264):

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 5184

The 36 conjugacy class representatives for

S_3\wr C_4

Character table for

S_3\wr C_4

Intermediate fields

\Q(\sqrt{5})

\Q(\zeta_{20})^+

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

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 12 sibling:	data not computed
Degree 18 sibling:	data not computed
Degree 24 siblings:	data not computed
Degree 36 siblings:	data not computed
Minimal sibling:	12.2.9642838578098750000.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }$	R	${\href{/padicField/7.4.0.1}{4} }^{3}$	${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$	${\href{/padicField/13.12.0.1}{12} }$	${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$	${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }$	${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.6.0.1}{6} }^{2}$	${\href{/padicField/37.4.0.1}{4} }^{3}$	${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$	${\href{/padicField/43.4.0.1}{4} }^{3}$	${\href{/padicField/47.12.0.1}{12} }$	${\href{/padicField/53.4.0.1}{4} }^{3}$	${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_K

for

p=7

in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_K

for

p=7

in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_K

for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of

[e_i,f_i]

for the factorization of the ideal

p\mathcal{O}_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	2.6.2.12a1.2	$x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 8 x^{3} + x^{2} + 4 x + 5$	$2$	$6$	$12$	$C_{12}$	$[2]^{6}$
$5$ 5	5.1.4.3a1.1	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
$5$ 5	5.2.4.6a1.2	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 664 x^{4} + 704 x^{3} + 416 x^{2} + 128 x + 21$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$19759$ 19759	$\Q_{19759}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{19759}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{19759}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{19759}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$

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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	12.6.158072000000000.1

Degree	$12$
Signature	$[6, 3]$
Discriminant	$-1.581\times 10^{14}$
Root discriminant	$15.25$
Ramified primes	$2,5,19759$
Class number	$1$
Class group	trivial
Galois group	$S_3\wr C_4$ (as 12T264)

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 $a$ )