magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 24, 42, -205, 27, 560, -342, -675, 547, 389, -356, -115, 112, 17, -17, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*x + 1)
gp: K = bnfinit(x^17 - x^16 - 17*x^15 + 17*x^14 + 112*x^13 - 115*x^12 - 356*x^11 + 389*x^10 + 547*x^9 - 675*x^8 - 342*x^7 + 560*x^6 + 27*x^5 - 205*x^4 + 42*x^3 + 24*x^2 - 10*x + 1, 1)
Normalized defining polynomial
\( x^{17} - x^{16} - 17 x^{15} + 17 x^{14} + 112 x^{13} - 115 x^{12} - 356 x^{11} + 389 x^{10} + 547 x^{9} - 675 x^{8} - 342 x^{7} + 560 x^{6} + 27 x^{5} - 205 x^{4} + 42 x^{3} + 24 x^{2} - 10 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $17$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[13, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(2232673506822932495146063557=3^{3}\cdot 205542871\cdot 402308340646912121\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.62$ | 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: | $3, 205542871, 402308340646912121$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{21} a^{16} - \frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{10}{21} a^{13} + \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{10}{21} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{21} a^{5} + \frac{2}{21} a^{4} + \frac{8}{21} a^{3} - \frac{3}{7} a^{2} + \frac{1}{3} a - \frac{1}{7}$
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: | $14$ | 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: | \( 129255854.893 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{17}$ (as 17T10):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 355687428096000 |
| The 297 conjugacy class representatives for $S_{17}$ are not computed |
| Character table for $S_{17}$ 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 | ${\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.8.0.1}{8} }$ | R | ${\href{/LocalNumberField/5.13.0.1}{13} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ | ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $17$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $17$ | ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.3.2 | $x^{4} - 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 3.13.0.1 | $x^{13} - x + 1$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
| 205542871 | Data not computed | ||||||
| 402308340646912121 | Data not computed | ||||||