magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -15, 54, -144, 336, -723, 1431, -2526, 3903, -5226, 6049, -6046, 5211, -3858, 2436, -1296, 570, -201, 54, -10, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 54*x^18 - 201*x^17 + 570*x^16 - 1296*x^15 + 2436*x^14 - 3858*x^13 + 5211*x^12 - 6046*x^11 + 6049*x^10 - 5226*x^9 + 3903*x^8 - 2526*x^7 + 1431*x^6 - 723*x^5 + 336*x^4 - 144*x^3 + 54*x^2 - 15*x + 3)
gp: K = bnfinit(x^20 - 10*x^19 + 54*x^18 - 201*x^17 + 570*x^16 - 1296*x^15 + 2436*x^14 - 3858*x^13 + 5211*x^12 - 6046*x^11 + 6049*x^10 - 5226*x^9 + 3903*x^8 - 2526*x^7 + 1431*x^6 - 723*x^5 + 336*x^4 - 144*x^3 + 54*x^2 - 15*x + 3, 1)
Normalized defining polynomial
\( x^{20} - 10 x^{19} + 54 x^{18} - 201 x^{17} + 570 x^{16} - 1296 x^{15} + 2436 x^{14} - 3858 x^{13} + 5211 x^{12} - 6046 x^{11} + 6049 x^{10} - 5226 x^{9} + 3903 x^{8} - 2526 x^{7} + 1431 x^{6} - 723 x^{5} + 336 x^{4} - 144 x^{3} + 54 x^{2} - 15 x + 3 \)
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: | \(947500495953542851857093=3^{18}\cdot 43\cdot 109\cdot 211\cdot 223^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.81$ | 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: | $3, 43, 109, 211, 223$ | 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: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( a^{18} - 9 a^{17} + 45 a^{16} - 156 a^{15} + 412 a^{14} - 868 a^{13} + 1499 a^{12} - 2156 a^{11} + 2607 a^{10} - 2662 a^{9} + 2300 a^{8} - 1682 a^{7} + 1048 a^{6} - 565 a^{5} + 276 a^{4} - 126 a^{3} + 54 a^{2} - 18 a + 5 \) (order $6$) | 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: | \( 22364.6774993 \) (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 245760 |
| The 201 conjugacy class representatives for t20n887 are not computed |
| Character table for t20n887 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 5.3.18063.1, 10.0.978815907.1 |
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 | $20$ | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.10.9.2 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ |
| 3.10.9.2 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $[\ ]_{10}^{4}$ | |
| 43 | Data not computed | ||||||
| $109$ | $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{109}$ | $x + 6$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 109.2.1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.6.0.1 | $x^{6} - x + 11$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 109.6.0.1 | $x^{6} - x + 11$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 211 | Data not computed | ||||||
| 223 | Data not computed | ||||||