magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 11, -26, -99, 432, 194, -2310, 560, 6142, -2890, -9294, 5041, 8478, -4721, -4790, 2640, 1679, -905, -354, 186, 41, -21, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 21*x^21 + 41*x^20 + 186*x^19 - 354*x^18 - 905*x^17 + 1679*x^16 + 2640*x^15 - 4790*x^14 - 4721*x^13 + 8478*x^12 + 5041*x^11 - 9294*x^10 - 2890*x^9 + 6142*x^8 + 560*x^7 - 2310*x^6 + 194*x^5 + 432*x^4 - 99*x^3 - 26*x^2 + 11*x - 1)
gp: K = bnfinit(x^23 - 2*x^22 - 21*x^21 + 41*x^20 + 186*x^19 - 354*x^18 - 905*x^17 + 1679*x^16 + 2640*x^15 - 4790*x^14 - 4721*x^13 + 8478*x^12 + 5041*x^11 - 9294*x^10 - 2890*x^9 + 6142*x^8 + 560*x^7 - 2310*x^6 + 194*x^5 + 432*x^4 - 99*x^3 - 26*x^2 + 11*x - 1, 1)
Normalized defining polynomial
\( x^{23} - 2 x^{22} - 21 x^{21} + 41 x^{20} + 186 x^{19} - 354 x^{18} - 905 x^{17} + 1679 x^{16} + 2640 x^{15} - 4790 x^{14} - 4721 x^{13} + 8478 x^{12} + 5041 x^{11} - 9294 x^{10} - 2890 x^{9} + 6142 x^{8} + 560 x^{7} - 2310 x^{6} + 194 x^{5} + 432 x^{4} - 99 x^{3} - 26 x^{2} + 11 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $23$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[19, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(21162307113254285003534815511322160672790793=3^{2}\cdot 31\cdot 61\cdot 12200617224119\cdot 101917126957405185189830413\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $76.51$ | 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: | $3, 31, 61, 12200617224119, 101917126957405185189830413$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
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: | $20$ | 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: | \( 288525695460000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{23}$ (as 23T7):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 25852016738884976640000 |
| The 1255 conjugacy class representatives for $S_{23}$ are not computed |
| Character table for $S_{23}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 46 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 | $15{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $23$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ | ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | R | $15{,}\,{\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $16{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 3.12.0.1 | $x^{12} - x^{4} - x^{3} - x^{2} + x - 1$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 31 | Data not computed | ||||||
| 61 | Data not computed | ||||||
| 12200617224119 | Data not computed | ||||||
| 101917126957405185189830413 | Data not computed | ||||||