magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 24, 9, -48, -47, 45, 54, -16, -39, 6, 13, 0, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 4*x^12 + 13*x^10 + 6*x^9 - 39*x^8 - 16*x^7 + 54*x^6 + 45*x^5 - 47*x^4 - 48*x^3 + 9*x^2 + 24*x + 4)
gp: K = bnfinit(x^13 - 4*x^12 + 13*x^10 + 6*x^9 - 39*x^8 - 16*x^7 + 54*x^6 + 45*x^5 - 47*x^4 - 48*x^3 + 9*x^2 + 24*x + 4, 1)
Normalized defining polynomial
\( x^{13} - 4 x^{12} + 13 x^{10} + 6 x^{9} - 39 x^{8} - 16 x^{7} + 54 x^{6} + 45 x^{5} - 47 x^{4} - 48 x^{3} + 9 x^{2} + 24 x + 4 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $13$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[5, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(5020153876560896=2^{10}\cdot 83\cdot 173\cdot 341423081\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.13$ | 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: | $2, 83, 173, 341423081$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$
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: | \( 1981.72650066 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{13}$ (as 13T9):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 6227020800 |
| The 101 conjugacy class representatives for $S_{13}$ are not computed |
| Character table for $S_{13}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 26 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 | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ | ${\href{/LocalNumberField/5.13.0.1}{13} }$ | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/31.13.0.1}{13} }$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.13.0.1}{13} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.4.8.2 | $x^{4} + 6 x^{2} + 1$ | $4$ | $1$ | $8$ | $C_2^2$ | $[2, 3]$ | |
| 2.7.0.1 | $x^{7} - x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 83 | Data not computed | ||||||
| $173$ | 173.2.1.2 | $x^{2} + 346$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 173.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 173.9.0.1 | $x^{9} - x + 8$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
| 341423081 | Data not computed | ||||||