magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-68, -24, 22, -32, 36, 40, -10, 8, 2, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 2*x^8 + 8*x^7 - 10*x^6 + 40*x^5 + 36*x^4 - 32*x^3 + 22*x^2 - 24*x - 68)
gp: K = bnfinit(x^10 - 4*x^9 + 2*x^8 + 8*x^7 - 10*x^6 + 40*x^5 + 36*x^4 - 32*x^3 + 22*x^2 - 24*x - 68, 1)
Normalized defining polynomial
\( x^{10} - 4 x^{9} + 2 x^{8} + 8 x^{7} - 10 x^{6} + 40 x^{5} + 36 x^{4} - 32 x^{3} + 22 x^{2} - 24 x - 68 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $10$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(3820118932979712=2^{30}\cdot 3^{5}\cdot 11^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.16$ | 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: | $2, 3, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{82498} a^{9} - \frac{5283}{41249} a^{8} - \frac{10850}{41249} a^{7} + \frac{614}{3173} a^{6} + \frac{7067}{41249} a^{5} + \frac{19056}{41249} a^{4} - \frac{15583}{41249} a^{3} + \frac{4120}{41249} a^{2} + \frac{2266}{41249} a - \frac{9084}{41249}$
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: | $5$ | 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: | \( 45699.9611546 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$\PGL(2,9)$ (as 10T30):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 720 |
| The 11 conjugacy class representatives for $\PGL(2,9)$ |
| Character table for $\PGL(2,9)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 sibling: | data not computed |
| Degree 45 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 | R | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.2 | $x^{2} + 6$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.8.27.84 | $x^{8} + 8 x^{5} + 10 x^{4} + 30$ | $8$ | $1$ | $27$ | $D_{8}$ | $[2, 3, 7/2, 9/2]$ | |
| $3$ | 3.10.5.1 | $x^{10} - 18 x^{6} + 81 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |