magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -3, 0, 0, 0, 0, 0, 0, 0, 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^27 - 3*x - 5)
gp: K = bnfinit(x^27 - 3*x - 5, 1)
Normalized defining polynomial
\( x^{27} - 3 x - 5 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $27$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 13]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-660756910949039852309163136206345937175666915026128585963=-\,3^{27}\cdot 7\cdot 139\cdot 373\cdot 857\cdot 14813\cdot 4501099\cdot 23278727\cdot 179491764750951217\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $127.19$ | 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: | $3, 7, 139, 373, 857, 14813, 4501099, 23278727, 179491764750951217$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$
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: | $13$ | 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: | \( 680758079489720300 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{27}$ (as 27T2392):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 10888869450418352160768000000 |
| The 3010 conjugacy class representatives for $S_{27}$ are not computed |
| Character table for $S_{27}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $23{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{6}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | R | $25{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | $21{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $17{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.13.0.1}{13} }{,}\,{\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/LocalNumberField/29.11.0.1}{11} }$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | $17{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ | $22{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $16{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 | Data not computed | ||||||
| $7$ | $\Q_{7}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.5.0.1 | $x^{5} - x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 7.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.12.0.1 | $x^{12} + 3 x^{2} - 2 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $139$ | 139.2.1.1 | $x^{2} - 139$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 139.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 139.10.0.1 | $x^{10} - x + 104$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 139.11.0.1 | $x^{11} - x + 24$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
| 373 | Data not computed | ||||||
| 857 | Data not computed | ||||||
| 14813 | Data not computed | ||||||
| 4501099 | Data not computed | ||||||
| 23278727 | Data not computed | ||||||
| 179491764750951217 | Data not computed | ||||||