magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 7, -24, 69, -172, 343, -523, 538, -166, -680, 1787, -2704, 3029, -2682, 1920, -1116, 522, -192, 53, -10, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 53*x^18 - 192*x^17 + 522*x^16 - 1116*x^15 + 1920*x^14 - 2682*x^13 + 3029*x^12 - 2704*x^11 + 1787*x^10 - 680*x^9 - 166*x^8 + 538*x^7 - 523*x^6 + 343*x^5 - 172*x^4 + 69*x^3 - 24*x^2 + 7*x - 1)
gp: K = bnfinit(x^20 - 10*x^19 + 53*x^18 - 192*x^17 + 522*x^16 - 1116*x^15 + 1920*x^14 - 2682*x^13 + 3029*x^12 - 2704*x^11 + 1787*x^10 - 680*x^9 - 166*x^8 + 538*x^7 - 523*x^6 + 343*x^5 - 172*x^4 + 69*x^3 - 24*x^2 + 7*x - 1, 1)
Normalized defining polynomial
\( x^{20} - 10 x^{19} + 53 x^{18} - 192 x^{17} + 522 x^{16} - 1116 x^{15} + 1920 x^{14} - 2682 x^{13} + 3029 x^{12} - 2704 x^{11} + 1787 x^{10} - 680 x^{9} - 166 x^{8} + 538 x^{7} - 523 x^{6} + 343 x^{5} - 172 x^{4} + 69 x^{3} - 24 x^{2} + 7 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(135802362058317231106057=47^{8}\cdot 269^{3}\cdot 293\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.34$ | 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: | $47, 269, 293$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
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: | $11$ | 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: | \( 3147.1828191 \) (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 solvable group of order 327680 |
| The 512 conjugacy class representatives for t20n905 are not computed |
| Character table for t20n905 is not computed |
Intermediate fields
| 5.1.2209.1, 10.2.1312634189.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ | $20$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| $47$ | $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{47}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 47.2.0.1 | $x^{2} - x + 13$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.4.2.1 | $x^{4} + 1175 x^{2} + 373321$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 1175 x^{2} + 373321$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.8.4.1 | $x^{8} + 172302 x^{4} - 103823 x^{2} + 7421994801$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 269 | Data not computed | ||||||
| 293 | Data not computed | ||||||