magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -3, -5, -2, 10, 18, 18, 2, -19, -27, -19, 2, 18, 18, 10, -2, -5, -3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 3*x^18 - 5*x^17 - 2*x^16 + 10*x^15 + 18*x^14 + 18*x^13 + 2*x^12 - 19*x^11 - 27*x^10 - 19*x^9 + 2*x^8 + 18*x^7 + 18*x^6 + 10*x^5 - 2*x^4 - 5*x^3 - 3*x^2 - x + 1)
gp: K = bnfinit(x^20 - x^19 - 3*x^18 - 5*x^17 - 2*x^16 + 10*x^15 + 18*x^14 + 18*x^13 + 2*x^12 - 19*x^11 - 27*x^10 - 19*x^9 + 2*x^8 + 18*x^7 + 18*x^6 + 10*x^5 - 2*x^4 - 5*x^3 - 3*x^2 - x + 1, 1)
Normalized defining polynomial
\( x^{20} - x^{19} - 3 x^{18} - 5 x^{17} - 2 x^{16} + 10 x^{15} + 18 x^{14} + 18 x^{13} + 2 x^{12} - 19 x^{11} - 27 x^{10} - 19 x^{9} + 2 x^{8} + 18 x^{7} + 18 x^{6} + 10 x^{5} - 2 x^{4} - 5 x^{3} - 3 x^{2} - x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-6736516842429754982164739=-\,11\cdot 17^{4}\cdot 157^{2}\cdot 4153^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.43$ | 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: | $11, 17, 157, 4153$ | 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: | $12$ | 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: | \( 40667.7930212 \) (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 3932160 |
| The 506 conjugacy class representatives for t20n1015 are not computed |
| Character table for t20n1015 is not computed |
Intermediate fields
| 5.5.70601.1, 10.10.782566688557.1 |
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 | $20$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ | R | ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ | $20$ | $20$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.6.0.1 | $x^{6} + x^{2} - 2 x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 11.6.0.1 | $x^{6} + x^{2} - 2 x + 8$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| $17$ | 17.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 17.3.0.1 | $x^{3} - x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 17.6.0.1 | $x^{6} - x + 12$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 17.8.4.1 | $x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $157$ | 157.2.1.2 | $x^{2} + 785$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 157.2.1.2 | $x^{2} + 785$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 157.3.0.1 | $x^{3} - x + 15$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 157.3.0.1 | $x^{3} - x + 15$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 157.4.0.1 | $x^{4} - x + 15$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 157.6.0.1 | $x^{6} - x + 61$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 4153 | Data not computed | ||||||