LMFDB - Number field 9.1.56158593639367311360.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 + 486*x - 1296)

gp: K = bnfinit(y^9 + 486*y - 1296, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 + 486*x - 1296);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 + 486*x - 1296)

$x^{9} + 486x - 1296$ x^9 + 486*x - 1296

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$9$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[1, 4]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$56158593639367311360$ 56158593639367311360 $\medspace = 2^{30}\cdot 3^{21}\cdot 5$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$156.45$	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:	$2^{137/32}3^{133/54}5^{1/2}\approx 650.7312305323685$
Ramified primes:	$2$ , $3$ , $5$ 2, 3, 5	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{15})$
$\Aut(K/\Q)$ :	$C_1$	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 $a$ )

$1$ , $a$ , $a^{2}$ , $\frac{1}{3}a^{3}$ , $\frac{1}{3}a^{4}$ , $\frac{1}{9}a^{5}$ , $\frac{1}{9}a^{6}$ , $\frac{1}{27}a^{7}$ , $\frac{1}{378}a^{8}-\frac{2}{189}a^{7}-\frac{2}{63}a^{6}+\frac{1}{63}a^{5}+\frac{1}{21}a^{4}+\frac{1}{7}a^{3}+\frac{3}{7}a^{2}+\frac{2}{7}a+\frac{1}{7}$ 1, a, a^2, 1/3*a^3, 1/3*a^4, 1/9*a^5, 1/9*a^6, 1/27*a^7, 1/378*a^8 - 2/189*a^7 - 2/63*a^6 + 1/63*a^5 + 1/21*a^4 + 1/7*a^3 + 3/7*a^2 + 2/7*a + 1/7

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 $1$ (assuming GRH) sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); oscar: class_group(K)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$4$	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:	$\frac{3067}{189}a^{8}+\frac{145}{21}a^{7}-\frac{6521}{63}a^{6}+\frac{11419}{63}a^{5}+\frac{1263}{7}a^{4}-\frac{28582}{21}a^{3}+\frac{13040}{7}a^{2}+\frac{24721}{7}a-\frac{68017}{7}$ , $\frac{731572}{9}a^{8}+\frac{331264}{9}a^{7}-\frac{5596384}{9}a^{6}+\frac{22935920}{9}a^{5}-7786976a^{4}+\frac{60399080}{3}a^{3}-45296225a^{2}+87155215a-91187647$ , $\frac{74467640890}{189}a^{8}-\frac{63141053977}{63}a^{7}+\frac{15532133031}{7}a^{6}-\frac{263286840812}{63}a^{5}+\frac{126046262893}{21}a^{4}-\frac{21142962682}{7}a^{3}-\frac{143387979517}{7}a^{2}+\frac{754872510668}{7}a-\frac{1196333843609}{7}$ , $\frac{21505669344580}{189}a^{8}+\frac{13791164409842}{63}a^{7}+\frac{2948001184686}{7}a^{6}+\frac{51043375723027}{63}a^{5}+\frac{32733116965789}{21}a^{4}+\frac{20991106478316}{7}a^{3}+\frac{40383556269062}{7}a^{2}+\frac{77691549805118}{7}a+\frac{536568255249847}{7}$ 3067/189a^8 + 145/21a^7 - 6521/63a^6 + 11419/63a^5 + 1263/7a^4 - 28582/21a^3 + 13040/7a^2 + 24721/7a - 68017/7, 731572/9a^8 + 331264/9a^7 - 5596384/9a^6 + 22935920/9a^5 - 7786976a^4 + 60399080/3a^3 - 45296225a^2 + 87155215a - 91187647, 74467640890/189a^8 - 63141053977/63a^7 + 15532133031/7a^6 - 263286840812/63a^5 + 126046262893/21a^4 - 21142962682/7a^3 - 143387979517/7a^2 + 754872510668/7a - 1196333843609/7, 21505669344580/189a^8 + 13791164409842/63a^7 + 2948001184686/7a^6 + 51043375723027/63a^5 + 32733116965789/21a^4 + 20991106478316/7a^3 + 40383556269062/7a^2 + 77691549805118/7a + 536568255249847/7 (assuming GRH)	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:	$7578466.00751$ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 7578466.00751 \cdot 1}{2\cdot\sqrt{56158593639367311360}}\cr\approx \mathstrut & 1.57613229819 \end{aligned}$ (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^9 + 486*x - 1296)

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^9 + 486*x - 1296, 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^9 + 486*x - 1296);

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^9 + 486*x - 1296);

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

$S_9$ (as 9T34):

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 362880

The 30 conjugacy class representatives for

S_9

Character table for

S_9

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

Degree 18 sibling:	data not computed
Degree 36 sibling:	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	R	R	${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$	${\href{/padicField/11.9.0.1}{9} }$	${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.9.0.1}{9} }$	${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.2.0.1}{2} }$	${\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.9.0.1}{9} }$	${\href{/padicField/59.9.0.1}{9} }$

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	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
$2$ 2	2.1.8.30a1.121	$x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 10$	$8$	$1$	$30$	$(((C_4 \times C_2): C_2):C_2):C_2$	$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$
$3$ 3	3.1.9.21b2.24	$x^{9} + 18 x^{6} + 9 x^{5} + 9 x^{4} + 18 x^{3} + 3$	$9$	$1$	$21$	$(C_3^2:C_3):C_2$	$[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$
$5$ 5	5.1.2.1a1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$5$ 5	5.7.1.0a1.1	$x^{7} + 3 x + 3$	$1$	$7$	$0$	$C_7$	$[\ ]^{7}$

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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	9.1.561...360.1

Degree	$9$
Signature	$[1, 4]$
Discriminant	$5.616\times 10^{19}$
Root discriminant	$156.45$
Ramified primes	$2,3,5$
Class number	$1$ (GRH)
Class group	trivial (GRH)
Galois group	$S_9$ (as 9T34)

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 $a$ )