magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17623, 5545, 842, 2173, 529, 71, 50, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - x^7 + 50*x^6 + 71*x^5 + 529*x^4 + 2173*x^3 + 842*x^2 + 5545*x + 17623)
gp: K = bnfinit(x^8 - x^7 + 50*x^6 + 71*x^5 + 529*x^4 + 2173*x^3 + 842*x^2 + 5545*x + 17623, 1)
Normalized defining polynomial
\( x^{8} - x^{7} + 50 x^{6} + 71 x^{5} + 529 x^{4} + 2173 x^{3} + 842 x^{2} + 5545 x + 17623 \)
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: | \(151939915084881=3^{6}\cdot 7^{6}\cdot 11^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $59.25$ | 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: | $3, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17} a^{6} + \frac{5}{17} a^{5} + \frac{3}{17} a^{4} - \frac{7}{17} a^{3} + \frac{1}{17} a^{2} - \frac{2}{17} a + \frac{5}{17}$, $\frac{1}{36270630857} a^{7} + \frac{374998582}{36270630857} a^{6} + \frac{17017736938}{36270630857} a^{5} + \frac{14057084747}{36270630857} a^{4} - \frac{15124706705}{36270630857} a^{3} + \frac{4190455030}{36270630857} a^{2} - \frac{10103876566}{36270630857} a - \frac{12323908292}{36270630857}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{6}\times C_{6}$, which has order $72$
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: | \( 104.827448547 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 8 |
| The 5 conjugacy class representatives for $Q_8$ |
| Character table for $Q_8$ |
Intermediate fields
| \(\Q(\sqrt{33}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{77}) \), \(\Q(\sqrt{21}, \sqrt{33})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | R | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.8.6.1 | $x^{8} + 9 x^{4} + 36$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| $7$ | 7.8.6.1 | $x^{8} + 35 x^{4} + 441$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
| $11$ | 11.8.6.1 | $x^{8} + 143 x^{4} + 5929$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.7_11.2t1.1c1 | $1$ | $ 7 \cdot 11 $ | $x^{2} - x - 19$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.3_11.2t1.1c1 | $1$ | $ 3 \cdot 11 $ | $x^{2} - x - 8$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.3_7.2t1.1c1 | $1$ | $ 3 \cdot 7 $ | $x^{2} - x - 5$ | $C_2$ (as 2T1) | $1$ | $1$ |
| *2 | 2.3e2_7e2_11e2.8t5.2c1 | $2$ | $ 3^{2} \cdot 7^{2} \cdot 11^{2}$ | $x^{8} - x^{7} + 50 x^{6} + 71 x^{5} + 529 x^{4} + 2173 x^{3} + 842 x^{2} + 5545 x + 17623$ | $Q_8$ (as 8T5) | $-1$ | $-2$ |
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 *.