Properties

Label 9.1.514147280633856.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{14}\cdot 3^{22}$
Root discriminant $43.11$
Ramified primes $2, 3$
Class number $7$
Class group $[7]$
Galois Group $\PSL(2,8)$ (as 9T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-128, -27, 36, 0, 0, -18, -12, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 12*x^6 - 18*x^5 + 36*x^2 - 27*x - 128)
gp: K = bnfinit(x^9 - 12*x^6 - 18*x^5 + 36*x^2 - 27*x - 128, 1)

Normalized defining polynomial

\(x^{9} \) \(\mathstrut -\mathstrut 12 x^{6} \) \(\mathstrut -\mathstrut 18 x^{5} \) \(\mathstrut +\mathstrut 36 x^{2} \) \(\mathstrut -\mathstrut 27 x \) \(\mathstrut -\mathstrut 128 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $9$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 4]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(514147280633856=2^{14}\cdot 3^{22}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $43.11$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{11468} a^{8} - \frac{779}{11468} a^{7} - \frac{963}{11468} a^{6} - \frac{989}{11468} a^{5} + \frac{2057}{11468} a^{4} + \frac{3117}{11468} a^{3} + \frac{3073}{11468} a^{2} - \frac{2753}{11468} a + \frac{11}{2867}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

$C_{7}$, which has order $7$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $4$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 3190.27571115 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$\PSL(2,8)$ (as 9T27):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 504
The 9 conjugacy class representatives for $\PSL(2,8)$
Character table for $\PSL(2,8)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 28 sibling: data not computed
Degree 36 sibling: 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 R R ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.14.6$x^{8} + 4 x^{7} + 4$$8$$1$$14$$C_2^3:C_7$$[2, 2, 2]^{7}$
$3$3.9.22.10$x^{9} + 6 x^{6} + 18 x^{5} + 6$$9$$1$$22$$D_{9}$$[2, 3]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
7.2e14_3e18.56.1c1$7$ $ 2^{14} \cdot 3^{18}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $-1$
7.2e14_3e16.56.1c1$7$ $ 2^{14} \cdot 3^{16}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $-1$
7.2e14_3e16.56.1c2$7$ $ 2^{14} \cdot 3^{16}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $-1$
7.2e14_3e16.56.1c3$7$ $ 2^{14} \cdot 3^{16}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $-1$
* 8.2e14_3e22.9t27.1c1$8$ $ 2^{14} \cdot 3^{22}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $0$
9.2e14_3e22.28t70.1c1$9$ $ 2^{14} \cdot 3^{22}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $1$
9.2e14_3e22.28t70.1c2$9$ $ 2^{14} \cdot 3^{22}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $1$
9.2e14_3e22.28t70.1c3$9$ $ 2^{14} \cdot 3^{22}$ $x^{9} - 12 x^{6} - 18 x^{5} + 36 x^{2} - 27 x - 128$ $\PSL(2,8)$ (as 9T27) $1$ $1$

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 *.