magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -6, 12, 115, -17, -847, -283, 3140, 1539, -6534, -3423, 8142, 4080, -6321, -2833, 3122, 1173, -982, -284, 191, 37, -21, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 21*x^21 + 37*x^20 + 191*x^19 - 284*x^18 - 982*x^17 + 1173*x^16 + 3122*x^15 - 2833*x^14 - 6321*x^13 + 4080*x^12 + 8142*x^11 - 3423*x^10 - 6534*x^9 + 1539*x^8 + 3140*x^7 - 283*x^6 - 847*x^5 - 17*x^4 + 115*x^3 + 12*x^2 - 6*x - 1)
gp: K = bnfinit(x^23 - 2*x^22 - 21*x^21 + 37*x^20 + 191*x^19 - 284*x^18 - 982*x^17 + 1173*x^16 + 3122*x^15 - 2833*x^14 - 6321*x^13 + 4080*x^12 + 8142*x^11 - 3423*x^10 - 6534*x^9 + 1539*x^8 + 3140*x^7 - 283*x^6 - 847*x^5 - 17*x^4 + 115*x^3 + 12*x^2 - 6*x - 1, 1)
Normalized defining polynomial
\( x^{23} - 2 x^{22} - 21 x^{21} + 37 x^{20} + 191 x^{19} - 284 x^{18} - 982 x^{17} + 1173 x^{16} + 3122 x^{15} - 2833 x^{14} - 6321 x^{13} + 4080 x^{12} + 8142 x^{11} - 3423 x^{10} - 6534 x^{9} + 1539 x^{8} + 3140 x^{7} - 283 x^{6} - 847 x^{5} - 17 x^{4} + 115 x^{3} + 12 x^{2} - 6 x - 1 \)
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: | \(14377559532718296764738220607061160634463664772=2^{2}\cdot 3594389883179574191184555151765290158615916193\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $101.59$ | 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: | $2, 3594389883179574191184555151765290158615916193$ | 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: | \( 9532691492330000 \) (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 | R | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | $22{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | $15{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ | ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | $20{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $19{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.11.0.1}{11} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.3.0.1 | $x^{3} - x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.12.0.1 | $x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| 3594389883179574191184555151765290158615916193 | Data not computed | ||||||