magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![321, 63, 313, -89, 100, -35, 17, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 3*x^7 + 17*x^6 - 35*x^5 + 100*x^4 - 89*x^3 + 313*x^2 + 63*x + 321)
gp: K = bnfinit(x^8 - 3*x^7 + 17*x^6 - 35*x^5 + 100*x^4 - 89*x^3 + 313*x^2 + 63*x + 321, 1)
Normalized defining polynomial
\( x^{8} - 3 x^{7} + 17 x^{6} - 35 x^{5} + 100 x^{4} - 89 x^{3} + 313 x^{2} + 63 x + 321 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(27485992521=3^{4}\cdot 13^{4}\cdot 109^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $20.18$ | 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: | $3, 13, 109$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{280} a^{6} - \frac{43}{280} a^{5} - \frac{5}{28} a^{4} + \frac{9}{20} a^{3} + \frac{13}{28} a^{2} - \frac{11}{280} a + \frac{3}{280}$, $\frac{1}{37520} a^{7} - \frac{23}{18760} a^{6} - \frac{5241}{37520} a^{5} + \frac{2799}{9380} a^{4} - \frac{13}{70} a^{3} - \frac{11881}{37520} a^{2} - \frac{4191}{9380} a + \frac{12591}{37520}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{4}$, which has order $4$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $3$ | 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: | \( 34.5772002628 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\wr C_2^2$ (as 8T29):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 64 |
| The 16 conjugacy class representatives for $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Character table for $(((C_4 \times C_2): C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.2.507.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| $109$ | 109.4.0.1 | $x^{4} - x + 30$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 109.4.2.2 | $x^{4} - 109 x^{2} + 71286$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.3_13_109.2t1.1c1 | $1$ | $ 3 \cdot 13 \cdot 109 $ | $x^{2} - x + 1063$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3_13.2t1.1c1 | $1$ | $ 3 \cdot 13 $ | $x^{2} - x + 10$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.109.2t1.1c1 | $1$ | $ 109 $ | $x^{2} - x - 27$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| * | 1.13.2t1.1c1 | $1$ | $ 13 $ | $x^{2} - x - 3$ | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.3_109.2t1.1c1 | $1$ | $ 3 \cdot 109 $ | $x^{2} - x + 82$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.3.2t1.1c1 | $1$ | $ 3 $ | $x^{2} - x + 1$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.13_109.2t1.1c1 | $1$ | $ 13 \cdot 109 $ | $x^{2} - x - 354$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.3_13e2_109.4t3.2c1 | $2$ | $ 3 \cdot 13^{2} \cdot 109 $ | $x^{4} - x^{3} - 61 x^{2} + 55 x + 1153$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.3_109.4t3.2c1 | $2$ | $ 3 \cdot 109 $ | $x^{4} - x^{3} - 4 x^{2} + 4 x + 7$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.3_13_109.4t3.3c1 | $2$ | $ 3 \cdot 13 \cdot 109 $ | $x^{4} - 2 x^{3} - 17 x^{2} + 18 x + 84$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.3_13_109e2.4t3.1c1 | $2$ | $ 3 \cdot 13 \cdot 109^{2}$ | $x^{4} - x^{3} + 26 x^{2} - 190 x - 2159$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| 2.3_13_109.4t3.4c1 | $2$ | $ 3 \cdot 13 \cdot 109 $ | $x^{4} - 2 x^{3} + 19 x^{2} - 18 x + 93$ | $D_{4}$ (as 4T3) | $1$ | $0$ | |
| * | 2.3_13.4t3.1c1 | $2$ | $ 3 \cdot 13 $ | $x^{4} - x^{3} - x^{2} - x + 1$ | $D_{4}$ (as 4T3) | $1$ | $0$ |
| * | 4.3e3_13e2_109e2.8t31.1c1 | $4$ | $ 3^{3} \cdot 13^{2} \cdot 109^{2}$ | $x^{8} - 3 x^{7} + 17 x^{6} - 35 x^{5} + 100 x^{4} - 89 x^{3} + 313 x^{2} + 63 x + 321$ | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $-2$ |
| 4.3_13e2_109e2.8t29.1c1 | $4$ | $ 3 \cdot 13^{2} \cdot 109^{2}$ | $x^{8} - 3 x^{7} + 17 x^{6} - 35 x^{5} + 100 x^{4} - 89 x^{3} + 313 x^{2} + 63 x + 321$ | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $2$ |
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 *.