magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 9*x - 9)
gp: K = bnfinit(x^21 + 9*x - 9, 1)
Normalized defining polynomial
\( x^{21} + 9 x - 9 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(23662383873250022452610353606982956821=3^{22}\cdot 37\cdot 139\cdot 193\cdot 223\cdot 3406529823078090997\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $60.22$ | 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: | $3, 37, 139, 193, 223, 3406529823078090997$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{3} a^{12}$, $\frac{1}{3} a^{13}$, $\frac{1}{3} a^{14}$, $\frac{1}{3} a^{15}$, $\frac{1}{3} a^{16}$, $\frac{1}{3} a^{17}$, $\frac{1}{3} a^{18}$, $\frac{1}{3} a^{19}$, $\frac{1}{3} a^{20}$
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: | \( 21551799245.1 \) (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 51090942171709440000 |
| The 792 conjugacy class representatives for S21 are not computed |
| Character table for S21 is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 42 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 | ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ | R | $21$ | ${\href{/LocalNumberField/7.11.0.1}{11} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $17{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ | ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | R | $20{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }$ | ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $37$ | $\Q_{37}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 37.2.1.2 | $x^{2} + 74$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 37.6.0.1 | $x^{6} - x + 20$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 37.12.0.1 | $x^{12} - x + 15$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $139$ | $\Q_{139}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 139.2.1.2 | $x^{2} + 556$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 139.5.0.1 | $x^{5} - x + 4$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 139.13.0.1 | $x^{13} - x + 5$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
| $193$ | 193.2.1.2 | $x^{2} + 965$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 193.3.0.1 | $x^{3} - x + 5$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 193.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 193.5.0.1 | $x^{5} - x + 5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 193.7.0.1 | $x^{7} - x + 17$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 223 | Data not computed | ||||||
| 3406529823078090997 | Data not computed | ||||||