magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -640, 140, 240, -360, -48, 150, 0, -20, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 20*x^8 + 150*x^6 - 48*x^5 - 360*x^4 + 240*x^3 + 140*x^2 - 640*x - 64)
gp: K = bnfinit(x^10 - 20*x^8 + 150*x^6 - 48*x^5 - 360*x^4 + 240*x^3 + 140*x^2 - 640*x - 64, 1)
Normalized defining polynomial
\( x^{10} - 20 x^{8} + 150 x^{6} - 48 x^{5} - 360 x^{4} + 240 x^{3} + 140 x^{2} - 640 x - 64 \)
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: | \(20480000000000000=2^{24}\cdot 5^{13}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $42.77$ | 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: | $2, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{65883472} a^{9} - \frac{1527035}{16470868} a^{8} - \frac{1922863}{16470868} a^{7} + \frac{991977}{4117717} a^{6} - \frac{2394621}{32941736} a^{5} + \frac{749034}{4117717} a^{4} - \frac{1885297}{8235434} a^{3} + \frac{621403}{4117717} a^{2} + \frac{1210791}{16470868} a + \frac{581747}{4117717}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}$, which has order $2$
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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 14846.6140672 \) | 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 | ${\href{/LocalNumberField/3.10.0.1}{10} }$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.10.0.1}{10} }$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.8.22.52 | $x^{8} + 2 x^{4} + 60$ | $4$ | $2$ | $22$ | $D_{8}$ | $[2, 3, 4]^{2}$ | |
| $5$ | 5.10.13.2 | $x^{10} + 10 x^{4} + 5$ | $10$ | $1$ | $13$ | $D_{10}$ | $[3/2]_{2}^{2}$ |