LMFDB - Number field 4.2.16560.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 + 4*x^2 - 12*x + 4)

gp: K = bnfinit(y^4 + 4*y^2 - 12*y + 4, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^4 + 4*x^2 - 12*x + 4);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^4 + 4*x^2 - 12*x + 4)

$x^{4} + 4x^{2} - 12x + 4$ x^4 + 4*x^2 - 12*x + 4

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$4$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[2, 1]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-16560$ -16560 $\medspace = -\,2^{4}\cdot 3^{2}\cdot 5\cdot 23$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$11.34$	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:	$2^{7/6}3^{1/2}5^{1/2}23^{1/2}\approx 41.697614426426426$
Ramified primes:	$2$ , $3$ , $5$ , $23$ 2, 3, 5, 23	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-115})$
$\Aut(K/\Q)$ :	$C_1$	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 $a$ )

$1$ , $a$ , $\frac{1}{2}a^{2}$ , $\frac{1}{2}a^{3}$ 1, a, 1/2*a^2, 1/2*a^3

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
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:	$2$	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:	$\frac{1}{2}a^{2}$ , $2a-3$ 1/2a^2, 2a - 3	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:	$12.092781486$	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 12.092781486 \cdot 1}{2\cdot\sqrt{16560}}\cr\approx \mathstrut & 1.1808803695 \end{aligned}$

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

x = polygen(QQ); K.<a> = NumberField(x^4 + 4*x^2 - 12*x + 4)

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^4 + 4*x^2 - 12*x + 4, 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^4 + 4*x^2 - 12*x + 4);

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^4 + 4*x^2 - 12*x + 4);

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

$S_4$ (as 4T5):

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 24

The 5 conjugacy class representatives for

S_4

Character table for

S_4

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and

\Q

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

Galois closure:	deg 24
Degree 6 siblings:	6.0.876024000.2, 6.2.7617600.2
Degree 8 sibling:	8.0.3626739360000.27
Degree 12 siblings:	deg 12, deg 12
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	R	R	${\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.4.0.1}{4} }$	${\href{/padicField/13.4.0.1}{4} }$	${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.4.0.1}{4} }$	R	${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.2.0.1}{2} }^{2}$	${\href{/padicField/47.4.0.1}{4} }$	${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

# 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	2.1.4.4a1.1	$x^{4} + 2 x + 2$	$4$	$1$	$4$	$S_4$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$
$3$ 3	3.1.2.1a1.2	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$3$ 3	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$5$ 5	5.2.1.0a1.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
$5$ 5	5.1.2.1a1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$23$ 23	23.2.1.0a1.1	$x^{2} + 21 x + 5$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
$23$ 23	23.1.2.1a1.1	$x^{2} + 23$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.115.2t1.a.a	$1$	$5 \cdot 23$	$\Q(\sqrt{-115})$	$C_2$ (as 2T1)	$1$	$-1$
	2.460.3t2.a.a	$2$	$2^{2} \cdot 5 \cdot 23$	3.1.460.1	$S_3$ (as 3T2)	$1$	$0$
	3.1904400.6t8.a.a	$3$	$2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}$	4.2.16560.1	$S_4$ (as 4T5)	$1$	$-1$
*	3.16560.4t5.a.a	$3$	$2^{4} \cdot 3^{2} \cdot 5 \cdot 23$	4.2.16560.1	$S_4$ (as 4T5)	$1$	$1$

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 *.

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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

Artin representations

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	4.2.16560.1

Degree	$4$
Signature	$[2, 1]$
Discriminant	$-16560$
Root discriminant	$11.34$
Ramified primes	$2,3,5,23$
Class number	$1$
Class group	trivial
Galois group	$S_4$ (as 4T5)

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 $a$ )