magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x - 1)
gp: K = bnfinit(x^20 - 8*x - 1, 1)
Normalized defining polynomial
\( x^{20} - 8 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-2124311929706157135254217773597696=-\,2^{10}\cdot 11443\cdot 20507\cdot 2525801\cdot 3500073554229929\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $46.38$ | 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: | $2, 11443, 20507, 2525801, 3500073554229929$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{10} + \frac{1}{4} a^{5} - \frac{1}{4}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{11} + \frac{1}{4} a^{6} - \frac{1}{4} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{12} + \frac{1}{4} a^{7} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{13} + \frac{1}{4} a^{8} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{14} + \frac{1}{4} a^{9} - \frac{1}{4} a^{4}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $10$ | 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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 783109824.543 \) (assuming GRH) | 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 non-solvable group of order 2432902008176640000 |
| The 627 conjugacy class representatives for t20n1117 are not computed |
| Character table for t20n1117 is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 40 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 | $15{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ | $17{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | $18{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ | $15{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | $20$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | $15{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }$ | ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 2.8.8.4 | $x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ | |
| 11443 | Data not computed | ||||||
| 20507 | Data not computed | ||||||
| 2525801 | Data not computed | ||||||
| 3500073554229929 | Data not computed | ||||||