magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![239, 0, 2519, 0, 9660, 0, 18633, 0, 20585, 0, 13915, 0, 5944, 0, 1610, 0, 268, 0, 25, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 25*x^18 + 268*x^16 + 1610*x^14 + 5944*x^12 + 13915*x^10 + 20585*x^8 + 18633*x^6 + 9660*x^4 + 2519*x^2 + 239)
gp: K = bnfinit(x^20 + 25*x^18 + 268*x^16 + 1610*x^14 + 5944*x^12 + 13915*x^10 + 20585*x^8 + 18633*x^6 + 9660*x^4 + 2519*x^2 + 239, 1)
Normalized defining polynomial
\( x^{20} + 25 x^{18} + 268 x^{16} + 1610 x^{14} + 5944 x^{12} + 13915 x^{10} + 20585 x^{8} + 18633 x^{6} + 9660 x^{4} + 2519 x^{2} + 239 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(11065929618893412926975840000000000000000=2^{20}\cdot 5^{16}\cdot 11^{4}\cdot 71^{4}\cdot 167^{4}\cdot 239\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $100.51$ | 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: | $2, 5, 11, 71, 167, 239$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{16} + \frac{2}{5} a^{14} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{19} - \frac{2}{5} a^{17} + \frac{2}{5} a^{15} + \frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{17902}$, which has order $35804$ (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: | $9$ | 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: | \( 29932844.3449 \) (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 1857945600 |
| The 260 conjugacy class representatives for t20n1106 are not computed |
| Character table for t20n1106 is not computed |
Intermediate fields
| 10.10.6645000909765625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ | R | $18{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | $16{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | Data not computed | ||||||
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.3.0.1 | $x^{3} - x + 2$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.5.8.7 | $x^{5} + 10 x^{4} + 5$ | $5$ | $1$ | $8$ | $F_5$ | $[2]^{4}$ | |
| 5.5.8.7 | $x^{5} + 10 x^{4} + 5$ | $5$ | $1$ | $8$ | $F_5$ | $[2]^{4}$ | |
| $11$ | 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 11.12.0.1 | $x^{12} - x + 7$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
| $71$ | 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 71.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 71.8.4.1 | $x^{8} + 110902 x^{4} - 357911 x^{2} + 3074813401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $167$ | $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{167}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 167.4.0.1 | $x^{4} - x + 60$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 167.6.0.1 | $x^{6} - x + 23$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 167.8.4.1 | $x^{8} + 3346680 x^{4} - 4657463 x^{2} + 2800066755600$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 239 | Data not computed | ||||||