magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -3, 3, 5, -8, -3, 13, -3, -11, 8, 6, -8, -1, 5, -1, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 1)
gp: K = bnfinit(x^17 - 2*x^16 - x^15 + 5*x^14 - x^13 - 8*x^12 + 6*x^11 + 8*x^10 - 11*x^9 - 3*x^8 + 13*x^7 - 3*x^6 - 8*x^5 + 5*x^4 + 3*x^3 - 3*x^2 + 1, 1)
Normalized defining polynomial
\( x^{17} - 2 x^{16} - x^{15} + 5 x^{14} - x^{13} - 8 x^{12} + 6 x^{11} + 8 x^{10} - 11 x^{9} - 3 x^{8} + 13 x^{7} - 3 x^{6} - 8 x^{5} + 5 x^{4} + 3 x^{3} - 3 x^{2} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $17$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(142063412679164600644=2^{2}\cdot 1697\cdot 20928611178427313\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.33$ | 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: | $2, 1697, 20928611178427313$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$
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: | $8$ | 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: | \( 1990.43257715 \) | 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 | R | $17$ | $17$ | ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ | $17$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ | ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.13.0.1}{13} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $17$ | ${\href{/LocalNumberField/47.13.0.1}{13} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.14.0.1 | $x^{14} - x^{5} - x^{3} - x + 1$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
| 1697 | Data not computed | ||||||
| 20928611178427313 | Data not computed | ||||||