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Properties

Label 9.1.736586891264.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{31}\cdot 7^{3}$
Root discriminant $20.82$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $(C_3^2:C_8):C_2$ (as 9T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 18, -32, 40, -40, 28, -16, 8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 8*x^7 - 16*x^6 + 28*x^5 - 40*x^4 + 40*x^3 - 32*x^2 + 18*x - 4)
 
gp: K = bnfinit(x^9 - 2*x^8 + 8*x^7 - 16*x^6 + 28*x^5 - 40*x^4 + 40*x^3 - 32*x^2 + 18*x - 4, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} + 8 x^{7} - 16 x^{6} + 28 x^{5} - 40 x^{4} + 40 x^{3} - 32 x^{2} + 18 x - 4 \)

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:  \(736586891264=2^{31}\cdot 7^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.82$
magma: Abs(Discriminant(Integers(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(Integers(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 \(a\))

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

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:  \( a^{8} - a^{7} + 7 a^{6} - 9 a^{5} + 19 a^{4} - 21 a^{3} + 19 a^{2} - 13 a + 5 \),  \( a^{8} - a^{7} + 7 a^{6} - 9 a^{5} + 19 a^{4} - 21 a^{3} + 18 a^{2} - 13 a + 3 \),  \( a^{8} + a^{7} + 6 a^{6} - 3 a^{5} + 9 a^{4} - 22 a^{3} + 20 a^{2} - 24 a + 17 \),  \( 46 a^{8} - 74 a^{7} + 339 a^{6} - 600 a^{5} + 1049 a^{4} - 1413 a^{3} + 1260 a^{2} - 944 a + 417 \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1103.79064185 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$PSU(3,2):C_2$ (as 9T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 9 conjugacy class representatives for $(C_3^2:C_8):C_2$
Character table for $(C_3^2:C_8):C_2$

Intermediate fields

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.8.31.173$x^{8} + 46$$8$$1$$31$$QD_{16}$$[2, 3, 4, 5]$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e2_7.2t1.1c1$1$ $ 2^{2} \cdot 7 $ $x^{2} - 7$ $C_2$ (as 2T1) $1$ $1$
1.2e3_7.2t1.1c1$1$ $ 2^{3} \cdot 7 $ $x^{2} - 14$ $C_2$ (as 2T1) $1$ $1$
2.2e8_7.4t3.5c1$2$ $ 2^{8} \cdot 7 $ $x^{4} - 8 x^{2} + 14$ $D_{4}$ (as 4T3) $1$ $2$
2.2e10_7.8t8.4c1$2$ $ 2^{10} \cdot 7 $ $x^{8} + 16 x^{6} + 80 x^{4} + 128 x^{2} + 56$ $QD_{16}$ (as 8T8) $0$ $-2$
2.2e10_7.8t8.4c2$2$ $ 2^{10} \cdot 7 $ $x^{8} + 16 x^{6} + 80 x^{4} + 128 x^{2} + 56$ $QD_{16}$ (as 8T8) $0$ $-2$
8.2e33_7e5.18t68.1c1$8$ $ 2^{33} \cdot 7^{5}$ $x^{9} - 2 x^{8} + 8 x^{7} - 16 x^{6} + 28 x^{5} - 40 x^{4} + 40 x^{3} - 32 x^{2} + 18 x - 4$ $(C_3^2:C_8):C_2$ (as 9T19) $1$ $0$
* 8.2e31_7e3.9t19.1c1$8$ $ 2^{31} \cdot 7^{3}$ $x^{9} - 2 x^{8} + 8 x^{7} - 16 x^{6} + 28 x^{5} - 40 x^{4} + 40 x^{3} - 32 x^{2} + 18 x - 4$ $(C_3^2:C_8):C_2$ (as 9T19) $1$ $0$

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 *.