Properties

Label 9.1.72313663744.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{8}\cdot 7^{10}$
Root discriminant $16.09$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois Group $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 26, 28, 0, 0, 0, -4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 4*x^7 + 28*x^3 + 26*x^2 + 9*x + 1)
gp: K = bnfinit(x^9 - x^8 - 4*x^7 + 28*x^3 + 26*x^2 + 9*x + 1, 1)

Normalized defining polynomial

\(x^{9} \) \(\mathstrut -\mathstrut x^{8} \) \(\mathstrut -\mathstrut 4 x^{7} \) \(\mathstrut +\mathstrut 28 x^{3} \) \(\mathstrut +\mathstrut 26 x^{2} \) \(\mathstrut +\mathstrut 9 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(72313663744=2^{8}\cdot 7^{10}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $16.09$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{758} a^{8} + \frac{130}{379} a^{7} - \frac{182}{379} a^{6} - \frac{127}{379} a^{5} - \frac{174}{379} a^{4} + \frac{66}{379} a^{3} + \frac{185}{379} a^{2} + \frac{165}{379} a - \frac{273}{758}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 295.021698724 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 1512
The 11 conjugacy class representatives for $\mathrm{P}\Gamma\mathrm{L}(2,8)$
Character table for $\mathrm{P}\Gamma\mathrm{L}(2,8)$

Intermediate fields

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

Sibling fields

Degree 27 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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$$0$Trivial$[\ ]$
2.8.8.13$x^{8} + 2 x + 2$$8$$1$$8$$C_2^3:(C_7: C_3)$$[8/7, 8/7, 8/7]_{7}^{3}$
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.7.10.5$x^{7} + 35 x^{4} + 7$$7$$1$$10$$F_7$$[5/3]_{3}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
7.2e8_7e10.56.1c1$7$ $ 2^{8} \cdot 7^{10}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $-1$
7.2e8_7e11.168.1c1$7$ $ 2^{8} \cdot 7^{11}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $-1$
7.2e8_7e11.168.1c2$7$ $ 2^{8} \cdot 7^{11}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $-1$
* 8.2e8_7e10.9t32.1c1$8$ $ 2^{8} \cdot 7^{10}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $0$
8.2e8_7e12.27t391.1c1$8$ $ 2^{8} \cdot 7^{12}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $0$
8.2e8_7e12.27t391.1c2$8$ $ 2^{8} \cdot 7^{12}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $0$ $0$
21.2e24_7e32.56.1c1$21$ $ 2^{24} \cdot 7^{32}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $-3$
27.2e30_7e42.28t165.1c1$27$ $ 2^{30} \cdot 7^{42}$ $x^{9} - x^{8} - 4 x^{7} + 28 x^{3} + 26 x^{2} + 9 x + 1$ $\mathrm{P}\Gamma\mathrm{L}(2,8)$ (as 9T32) $1$ $3$

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 *.