Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 2*x^7 - 2*x^5 + 2*x^4 - x + 1)
gp: K = bnfinit(x^9 - 2*x^8 + 2*x^7 - 2*x^5 + 2*x^4 - x + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 0, 0, 2, -2, 0, 2, -2, 1]);
\(x^{9} - 2 x^{8} + 2 x^{7} - 2 x^{5} + 2 x^{4} - x + 1\)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
Degree: | $9$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[1, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
Discriminant: | \(33860761\)\(\medspace = 11^{2}\cdot 23^{4}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $6.86$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
Ramified primes: | $11, 23$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
$|\Aut(K/\Q)|$: | $1$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
Torsion generator: | \( -1 \) (order $2$) ![]() | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
Fundamental units: | \( a \), \( a^{8} - a^{7} + a^{6} - a^{4} + a - 1 \), \( a^{8} - a^{7} + a^{6} - a^{4} + a^{3} + a \), \( a - 1 \) ![]() | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
Regulator: | \( 0.744153660637 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
A solvable group of order 648 |
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
Intermediate fields
3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | 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 | ${\href{/LocalNumberField/2.9.0.1}{9} }$ | ${\href{/LocalNumberField/3.9.0.1}{9} }$ | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/13.9.0.1}{9} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }$ | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.9.0.1}{9} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
$11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
11.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.4.2.2 | $x^{4} - 11 x^{2} + 847$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
$23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.4.3.1 | $x^{4} + 46$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.23.3t2.b.a | $2$ | $ 23 $ | 3.1.23.1 | $S_3$ (as 3T2) | $1$ | $0$ |
3.64009.6t8.a.a | $3$ | $ 11^{2} \cdot 23^{2}$ | 4.2.1472207.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.1472207.4t5.a.a | $3$ | $ 11^{2} \cdot 23^{3}$ | 4.2.1472207.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
6.778797503.18t217.a.a | $6$ | $ 11^{2} \cdot 23^{5}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
* | 6.1472207.9t30.a.a | $6$ | $ 11^{2} \cdot 23^{3}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ |
6.94234497863.36t1121.a.a | $6$ | $ 11^{4} \cdot 23^{5}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
6.94234497863.36t1121.a.b | $6$ | $ 11^{4} \cdot 23^{5}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.216...849.12t177.a.a | $8$ | $ 11^{4} \cdot 23^{6}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.216...849.12t177.b.a | $8$ | $ 11^{4} \cdot 23^{6}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
8.216...849.12t178.a.a | $8$ | $ 11^{4} \cdot 23^{6}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $0$ | |
12.319...743.36t1123.a.a | $12$ | $ 11^{6} \cdot 23^{9}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $-2$ | |
12.319...743.18t218.a.a | $12$ | $ 11^{6} \cdot 23^{9}$ | 9.1.33860761.1 | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) | $1$ | $2$ |
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 *.