magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 7, -16, 32, -59, 97, -146, 202, -258, 306, -335, 343, -325, 285, -230, 170, -114, 68, -35, 15, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*x + 1)
gp: K = bnfinit(x^22 - 5*x^21 + 15*x^20 - 35*x^19 + 68*x^18 - 114*x^17 + 170*x^16 - 230*x^15 + 285*x^14 - 325*x^13 + 343*x^12 - 335*x^11 + 306*x^10 - 258*x^9 + 202*x^8 - 146*x^7 + 97*x^6 - 59*x^5 + 32*x^4 - 16*x^3 + 7*x^2 - 2*x + 1, 1)
Normalized defining polynomial
\( x^{22} - 5 x^{21} + 15 x^{20} - 35 x^{19} + 68 x^{18} - 114 x^{17} + 170 x^{16} - 230 x^{15} + 285 x^{14} - 325 x^{13} + 343 x^{12} - 335 x^{11} + 306 x^{10} - 258 x^{9} + 202 x^{8} - 146 x^{7} + 97 x^{6} - 59 x^{5} + 32 x^{4} - 16 x^{3} + 7 x^{2} - 2 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-103946711888571010237690363=-\,79\cdot 22109\cdot 59513372976908430233\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.23$ | 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: | $79, 22109, 59513372976908430233$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{43} a^{21} + \frac{7}{43} a^{20} + \frac{13}{43} a^{19} - \frac{8}{43} a^{18} + \frac{15}{43} a^{17} - \frac{20}{43} a^{16} + \frac{16}{43} a^{15} + \frac{5}{43} a^{14} + \frac{1}{43} a^{13} - \frac{12}{43} a^{12} - \frac{16}{43} a^{11} - \frac{11}{43} a^{10} + \frac{2}{43} a^{9} - \frac{19}{43} a^{8} + \frac{17}{43} a^{7} + \frac{15}{43} a^{6} + \frac{19}{43} a^{5} - \frac{3}{43} a^{4} - \frac{4}{43} a^{3} - \frac{21}{43} a^{2} + \frac{13}{43} a - \frac{18}{43}$
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: | $10$ | 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: | \( 14889.9809115 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 1124000727777607680000 |
| The 1002 conjugacy class representatives for t22n59 are not computed |
| Character table for t22n59 is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 44 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 | ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | $15{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | $16{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | $15{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ | $22$ | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ | $22$ | ${\href{/LocalNumberField/53.13.0.1}{13} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $79$ | $\Q_{79}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 79.2.1.2 | $x^{2} + 158$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 79.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.3.0.1 | $x^{3} - x + 4$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 79.6.0.1 | $x^{6} - x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 79.8.0.1 | $x^{8} - x + 43$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 22109 | Data not computed | ||||||
| 59513372976908430233 | Data not computed | ||||||