LMFDB - Number field 6.0.647340405907.3

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928)

gp: K = bnfinit(y^6 - y^5 + 63*y^4 + 210*y^3 + 2460*y^2 + 5968*y + 36928, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928)

$x^{6} - x^{5} + 63x^{4} + 210x^{3} + 2460x^{2} + 5968x + 36928$ x^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$6$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 3]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-647340405907$ -647340405907 $\medspace = -\,13^{4}\cdot 283^{3}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$93.01$	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:	$13^{2/3}283^{1/2}\approx 93.00838841805972$
Ramified primes:	$13$ , $283$ 13, 283	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-283})$
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$ :	$S_3$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois over $\Q$ .
This is not a CM field.

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

$1$ , $a$ , $a^{2}$ , $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$ , $\frac{1}{12}a^{4}+\frac{1}{12}a^{3}-\frac{1}{4}a^{2}+\frac{1}{3}a-\frac{1}{3}$ , $\frac{1}{1538160}a^{5}+\frac{461}{512720}a^{4}+\frac{23923}{307632}a^{3}+\frac{2441}{5304}a^{2}+\frac{34169}{76908}a+\frac{21763}{96135}$ 1, a, a^2, 1/2*a^3 - 1/2*a^2 - 1/2*a, 1/12*a^4 + 1/12*a^3 - 1/4*a^2 + 1/3*a - 1/3, 1/1538160*a^5 + 461/512720*a^4 + 23923/307632*a^3 + 2441/5304*a^2 + 34169/76908*a + 21763/96135

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		$4$
Inessential primes:		$2$

Class group and class number

$C_{3}\times C_{3}\times C_{9}$ , which has order $81$

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:	$\frac{461}{256360}a^{5}-\frac{3337}{256360}a^{4}+\frac{5023}{51272}a^{3}-\frac{31}{884}a^{2}+\frac{8907}{6409}a+\frac{229631}{32045}$ , $\frac{7}{10608}a^{5}-\frac{43}{10608}a^{4}+\frac{157}{10608}a^{3}-\frac{679}{5304}a^{2}+\frac{1133}{884}a-\frac{14293}{663}$ 461/256360a^5 - 3337/256360a^4 + 5023/51272a^3 - 31/884a^2 + 8907/6409a + 229631/32045, 7/10608a^5 - 43/10608a^4 + 157/10608a^3 - 679/5304a^2 + 1133/884a - 14293/663	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:	$114.647869662$	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 114.647869662 \cdot 81}{2\cdot\sqrt{647340405907}}\cr\approx \mathstrut & 1.43150958292 \end{aligned}$

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

x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928)

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^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928, 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^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928);

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^6 - x^5 + 63*x^4 + 210*x^3 + 2460*x^2 + 5968*x + 36928);

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

$S_3$ (as 6T2):

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 6

The 3 conjugacy class representatives for

S_3

Character table for

S_3

Intermediate fields

\Q(\sqrt{-283})

, 3.1.47827.2 x3

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 algebras

Twin sextic algebra:	3.1.47827.2 $\times$ $\Q$ $\times$ $\Q$ $\times$ $\Q$
Degree 3 sibling:	3.1.47827.2
Minimal sibling:	3.1.47827.2

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.2.0.1}{2} }^{3}$	${\href{/padicField/3.2.0.1}{2} }^{3}$	${\href{/padicField/5.2.0.1}{2} }^{3}$	${\href{/padicField/7.3.0.1}{3} }^{2}$	${\href{/padicField/11.1.0.1}{1} }^{6}$	R	${\href{/padicField/17.2.0.1}{2} }^{3}$	${\href{/padicField/19.2.0.1}{2} }^{3}$	${\href{/padicField/23.3.0.1}{3} }^{2}$	${\href{/padicField/29.1.0.1}{1} }^{6}$	${\href{/padicField/31.2.0.1}{2} }^{3}$	${\href{/padicField/37.2.0.1}{2} }^{3}$	${\href{/padicField/41.1.0.1}{1} }^{6}$	${\href{/padicField/43.2.0.1}{2} }^{3}$	${\href{/padicField/47.2.0.1}{2} }^{3}$	${\href{/padicField/53.2.0.1}{2} }^{3}$	${\href{/padicField/59.3.0.1}{3} }^{2}$

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
$13$ 13	13.1.3.2a1.1	$x^{3} + 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$13$ 13	13.1.3.2a1.1	$x^{3} + 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$283$ 283		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		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 algebras

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	6.0.647340405907.3

Degree	$6$
Signature	$[0, 3]$
Discriminant	$-647340405907$
Root discriminant	$93.01$
Ramified primes	$13,283$
Class number	$81$
Class group	[3, 3, 9]
Galois group	$S_3$ (as 6T2)

Properties

Related objects

Downloads

Learn more

Integral basis (with respect to field generator aaa)

Class group and class number

Sibling algebras

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 $a$ )