magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-400, -500, -540, -250, -26, 73, 34, -8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 - 8*x^7 + 34*x^6 + 73*x^5 - 26*x^4 - 250*x^3 - 540*x^2 - 500*x - 400)
gp: K = bnfinit(x^9 - 4*x^8 - 8*x^7 + 34*x^6 + 73*x^5 - 26*x^4 - 250*x^3 - 540*x^2 - 500*x - 400, 1)
Normalized defining polynomial
\( x^{9} - 4 x^{8} - 8 x^{7} + 34 x^{6} + 73 x^{5} - 26 x^{4} - 250 x^{3} - 540 x^{2} - 500 x - 400 \)
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: | \(2379293284000000=2^{8}\cdot 5^{6}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.11$ | 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: | $2, 5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{5} + \frac{3}{10} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{30} a^{6} - \frac{1}{30} a^{5} - \frac{1}{15} a^{4} - \frac{7}{30} a^{3} + \frac{3}{10} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{90} a^{7} + \frac{1}{90} a^{6} + \frac{1}{45} a^{5} + \frac{2}{45} a^{4} + \frac{13}{90} a^{3} - \frac{31}{90} a^{2} + \frac{2}{9} a - \frac{1}{9}$, $\frac{1}{2700} a^{8} - \frac{1}{225} a^{7} - \frac{1}{1350} a^{6} - \frac{2}{135} a^{5} + \frac{71}{900} a^{4} - \frac{37}{135} a^{3} - \frac{1}{3} a^{2} + \frac{4}{27}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{3}$, which has order $3$
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: | \( 30238.329668 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$PSU(3,2)$ (as 9T14):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 72 |
| The 6 conjugacy class representatives for $C_3^2:Q_8$ |
| Character table for $C_3^2:Q_8$ |
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 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.4.4.2 | $x^{4} - x^{2} + 5$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.4.4.2 | $x^{4} - x^{2} + 5$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| $5$ | $\Q_{5}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $29$ | $\Q_{29}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 29.4.3.2 | $x^{4} - 116$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 29.4.3.2 | $x^{4} - 116$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.5.2t1.1c1 | $1$ | $ 5 $ | $x^{2} - x - 1$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.29.2t1.1c1 | $1$ | $ 29 $ | $x^{2} - x - 7$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.5_29.2t1.1c1 | $1$ | $ 5 \cdot 29 $ | $x^{2} - x - 36$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.2e4_5e2_29e2.8t5.1c1 | $2$ | $ 2^{4} \cdot 5^{2} \cdot 29^{2}$ | $x^{8} + 145 x^{6} + 6380 x^{4} + 105125 x^{2} + 525625$ | $Q_8$ (as 8T5) | $-1$ | $-2$ | |
| * | 8.2e8_5e6_29e6.9t14.1c1 | $8$ | $ 2^{8} \cdot 5^{6} \cdot 29^{6}$ | $x^{9} - 4 x^{8} - 8 x^{7} + 34 x^{6} + 73 x^{5} - 26 x^{4} - 250 x^{3} - 540 x^{2} - 500 x - 400$ | $C_3^2:Q_8$ (as 9T14) | $1$ | $0$ |
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 *.