magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![863, -544, -1302, 240, 457, 2, -41, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 - 41*x^6 + 2*x^5 + 457*x^4 + 240*x^3 - 1302*x^2 - 544*x + 863)
gp: K = bnfinit(x^8 - x^7 - 41*x^6 + 2*x^5 + 457*x^4 + 240*x^3 - 1302*x^2 - 544*x + 863, 1)
Normalized defining polynomial
\( x^{8} - x^{7} - 41 x^{6} + 2 x^{5} + 457 x^{4} + 240 x^{3} - 1302 x^{2} - 544 x + 863 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6134265795025=5^{2}\cdot 19^{2}\cdot 29^{4}\cdot 31^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.67$ | 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: | $5, 19, 29, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{39} a^{6} + \frac{16}{39} a^{5} - \frac{17}{39} a^{4} - \frac{4}{39} a^{3} + \frac{1}{13} a^{2} - \frac{4}{39} a - \frac{8}{39}$, $\frac{1}{4826055} a^{7} - \frac{7984}{965211} a^{6} + \frac{1834504}{4826055} a^{5} - \frac{1625329}{4826055} a^{4} - \frac{74034}{1608685} a^{3} + \frac{1988138}{4826055} a^{2} - \frac{2235059}{4826055} a - \frac{36512}{123745}$
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: | $7$ | 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: | \( 5814.836395889703 \) | 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
| \(\Q(\sqrt{29}) \), 4.4.4205.1 |
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 | ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | R | R | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.4.2.2 | $x^{4} - 19 x^{2} + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $29$ | 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $31$ | 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.5_19_29_31.2t1.1c1 | $1$ | $ 5 \cdot 19 \cdot 29 \cdot 31 $ | $x^{2} - x - 21351$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.19_29_31.2t1.1c1 | $1$ | $ 19 \cdot 29 \cdot 31 $ | $x^{2} - x - 4270$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 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_19_31.2t1.1c1 | $1$ | $ 5 \cdot 19 \cdot 31 $ | $x^{2} - x - 736$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.19_31.2t1.1c1 | $1$ | $ 19 \cdot 31 $ | $x^{2} - x - 147$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.5_29.2t1.1c1 | $1$ | $ 5 \cdot 29 $ | $x^{2} - x - 36$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 2.5_19_29_31.4t3.1c1 | $2$ | $ 5 \cdot 19 \cdot 29 \cdot 31 $ | $x^{4} - 2 x^{3} - 130 x^{2} + 131 x + 20$ | $D_{4}$ (as 4T3) | $1$ | $2$ | |
| 2.5_19_29_31.4t3.2c1 | $2$ | $ 5 \cdot 19 \cdot 29 \cdot 31 $ | $x^{4} - 2 x^{3} - 142 x^{2} + 143 x + 842$ | $D_{4}$ (as 4T3) | $1$ | $2$ | |
| 2.5_19e2_29_31e2.4t3.2c1 | $2$ | $ 5 \cdot 19^{2} \cdot 29 \cdot 31^{2}$ | $x^{4} - x^{3} - 1620 x^{2} + 442 x + 628279$ | $D_{4}$ | $1$ | $2$ | |
| 2.5_19_31.4t3.2c1 | $2$ | $ 5 \cdot 19 \cdot 31 $ | $x^{4} - 2 x^{3} - 24 x^{2} + 25 x + 9$ | $D_{4}$ (as 4T3) | $1$ | $2$ | |
| 2.5_19_29e2_31.4t3.1c1 | $2$ | $ 5 \cdot 19 \cdot 29^{2} \cdot 31 $ | $x^{4} - x^{3} - 414 x^{2} - 1928 x + 10607$ | $D_{4}$ (as 4T3) | $1$ | $2$ | |
| * | 2.5_29.4t3.2c1 | $2$ | $ 5 \cdot 29 $ | $x^{4} - x^{3} - 3 x^{2} + x + 1$ | $D_{4}$ (as 4T3) | $1$ | $2$ |
| * | 4.5_19e2_29e2_31e2.8t29.1c1 | $4$ | $ 5 \cdot 19^{2} \cdot 29^{2} \cdot 31^{2}$ | $x^{8} - x^{7} - 41 x^{6} + 2 x^{5} + 457 x^{4} + 240 x^{3} - 1302 x^{2} - 544 x + 863$ | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $4$ |
| 4.5e3_19e2_29e2_31e2.8t29.1c1 | $4$ | $ 5^{3} \cdot 19^{2} \cdot 29^{2} \cdot 31^{2}$ | $x^{8} - x^{7} - 41 x^{6} + 2 x^{5} + 457 x^{4} + 240 x^{3} - 1302 x^{2} - 544 x + 863$ | $(((C_4 \times C_2): C_2):C_2):C_2$ (as 8T29) | $1$ | $4$ |
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 *.