# Properties

 Label 9.3.962721767.1 Degree $9$ Signature $[3, 3]$ Discriminant $-962721767$ Root discriminant $$9.96$$ Ramified primes $7,167$ Class number $1$ Class group trivial Galois group $S_3 \wr C_3$ (as 9T28)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

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

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

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

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*x - 1)

$$x^{9} - x^{8} + 2x^{7} - 4x^{2} + 4x - 1$$

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: $[3, 3]$ sage: K.signature()  gp: K.sign  magma: Signature(K);  oscar: signature(K) Discriminant: $$-962721767$$ -962721767 $$\medspace = -\,7^{8}\cdot 167$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$9.96$$ 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)) Ramified primes: $$7$$, $$167$$ 7, 167 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{-167})$$ $\card{ \Aut(K/\Q) }$: $1$ 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 $$a$$)

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

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: $5$ 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: $a^{8}-a^{7}+2a^{6}-4a+4$, $9a^{8}-3a^{7}+16a^{6}+11a^{5}+7a^{4}+5a^{3}+4a^{2}-34a+14$, $a^{7}+2a^{5}+2a^{4}+2a^{3}+2a^{2}+a-2$, $a^{8}-a^{7}+2a^{6}-4a+3$, $8a^{8}-3a^{7}+14a^{6}+9a^{5}+5a^{4}+4a^{3}+2a^{2}-30a+13$ a^8 - a^7 + 2*a^6 - 4*a + 4, 9*a^8 - 3*a^7 + 16*a^6 + 11*a^5 + 7*a^4 + 5*a^3 + 4*a^2 - 34*a + 14, a^7 + 2*a^5 + 2*a^4 + 2*a^3 + 2*a^2 + a - 2, a^8 - a^7 + 2*a^6 - 4*a + 3, 8*a^8 - 3*a^7 + 14*a^6 + 9*a^5 + 5*a^4 + 4*a^3 + 2*a^2 - 30*a + 13 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: $$7.77137506101$$ 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^{3}\cdot(2\pi)^{3}\cdot 7.77137506101 \cdot 1}{2\cdot\sqrt{962721767}}\cr\approx \mathstrut & 0.248511831519 \end{aligned}

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

x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*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))))

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

K = bnfinit(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*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)))]

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

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*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)));

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

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 + 2*x^7 - 4*x^2 + 4*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

$S_3\wr C_3$ (as 9T28):

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 648 The 17 conjugacy class representatives for $S_3 \wr C_3$ Character table for $S_3 \wr C_3$

## Intermediate fields

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 siblings: data not computed Degree 24 siblings: data not computed Degree 27 siblings: data not computed Degree 36 siblings: data not computed

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ R ${\href{/padicField/11.3.0.1}{3} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.3.0.1}{3} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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$LabelPolynomial$efc$Galois group Slope content $$7$$ 7.9.8.2$x^{9} + 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3} $$167$$ \Q_{167}$$x + 162$$1$$1$$0Trivial[\ ] \Q_{167}$$x + 162$$1$$1$$0Trivial[\ ] \Q_{167}$$x + 162$$1$$1$$0Trivial[\ ] \Q_{167}$$x + 162$$1$$1$$0Trivial[\ ] \Q_{167}$$x + 162$$1$$1$$0Trivial[\ ] 167.2.0.1x^{2} + 166 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$167.2.1.1$x^{2} + 835$$2$$1$$1$$C_2[\ ]_{2}$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$1.167.2t1.a.a$1 167 $$$\Q(\sqrt{-167})$$$C_2$(as 2T1)$1-1$1.1169.6t1.b.a$1 7 \cdot 167 $6.0.11182568663.3$C_6$(as 6T1)$0-1$* 1.7.3t1.a.a$1 7 $$$\Q(\zeta_{7})^+$$$C_3$(as 3T1)$01$* 1.7.3t1.a.b$1 7 $$$\Q(\zeta_{7})^+$$$C_3$(as 3T1)$01$1.1169.6t1.b.b$1 7 \cdot 167 $6.0.11182568663.3$C_6$(as 6T1)$0-1$3.8183.6t6.a.a$3 7^{2} \cdot 167 $6.0.400967.1$A_4\times C_2$(as 6T6)$1-3$3.1366561.4t4.a.a$3 7^{2} \cdot 167^{2}$4.4.1366561.1$A_4$(as 4T4)$13$6.547945864487.18t197.a.a$6 7^{6} \cdot 167^{3}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$10$* 6.19647383.9t28.a.a$6 7^{6} \cdot 167 $9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$10$6.152...943.18t202.a.a$6 7^{6} \cdot 167^{5}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$10$6.547945864487.18t197.b.a$6 7^{6} \cdot 167^{3}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$10$8.448...121.12t176.a.a$8 7^{8} \cdot 167^{4}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$10$8.640...303.24t1539.a.a$8 7^{7} \cdot 167^{4}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$00$8.640...303.24t1539.a.b$8 7^{7} \cdot 167^{4}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$00$12.219...929.18t206.a.a$12 7^{10} \cdot 167^{4}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$10$12.170...209.36t1101.a.a$12 7^{10} \cdot 167^{8}$9.3.962721767.1$S_3 \wr C_3 $(as 9T28)$10\$

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 *.