magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 39, 19, -537, 356, 2503, -2661, -5405, 7285, 5920, -10393, -2935, 8563, -105, -4167, 862, 1148, -422, -150, 88, 2, -7, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 356*x^5 - 537*x^4 + 19*x^3 + 39*x^2 - 2*x - 1)
gp: K = bnfinit(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 356*x^5 - 537*x^4 + 19*x^3 + 39*x^2 - 2*x - 1, 1)
Normalized defining polynomial
\( x^{23} - 7 x^{22} + 2 x^{21} + 88 x^{20} - 150 x^{19} - 422 x^{18} + 1148 x^{17} + 862 x^{16} - 4167 x^{15} - 105 x^{14} + 8563 x^{13} - 2935 x^{12} - 10393 x^{11} + 5920 x^{10} + 7285 x^{9} - 5405 x^{8} - 2661 x^{7} + 2503 x^{6} + 356 x^{5} - 537 x^{4} + 19 x^{3} + 39 x^{2} - 2 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $23$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[13, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-425103981715298419969372518134528837127343=-\,13\cdot 1289\cdot 4057217\cdot 26391289\cdot 32505503\cdot 7288751994309941\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $64.56$ | 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: | $13, 1289, 4057217, 26391289, 32505503, 7288751994309941$ | 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: | $17$ | 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: | \( 4090939159600 \) (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 | ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ | $18{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ | $22{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | ${\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | $19{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | $20{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $13$ | 13.2.1.1 | $x^{2} - 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.8.0.1 | $x^{8} + 4 x^{2} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 13.13.0.1 | $x^{13} - x + 2$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
| 1289 | Data not computed | ||||||
| 4057217 | Data not computed | ||||||
| 26391289 | Data not computed | ||||||
| 32505503 | Data not computed | ||||||
| 7288751994309941 | Data not computed | ||||||