magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -20, -127, -184, 719, 1511, -1665, -4063, 2182, 5773, -1794, -4945, 927, 2669, -284, -906, 46, 186, -3, -21, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 46*x^16 - 906*x^15 - 284*x^14 + 2669*x^13 + 927*x^12 - 4945*x^11 - 1794*x^10 + 5773*x^9 + 2182*x^8 - 4063*x^7 - 1665*x^6 + 1511*x^5 + 719*x^4 - 184*x^3 - 127*x^2 - 20*x - 1)
gp: K = bnfinit(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 46*x^16 - 906*x^15 - 284*x^14 + 2669*x^13 + 927*x^12 - 4945*x^11 - 1794*x^10 + 5773*x^9 + 2182*x^8 - 4063*x^7 - 1665*x^6 + 1511*x^5 + 719*x^4 - 184*x^3 - 127*x^2 - 20*x - 1, 1)
Normalized defining polynomial
\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 46 x^{16} - 906 x^{15} - 284 x^{14} + 2669 x^{13} + 927 x^{12} - 4945 x^{11} - 1794 x^{10} + 5773 x^{9} + 2182 x^{8} - 4063 x^{7} - 1665 x^{6} + 1511 x^{5} + 719 x^{4} - 184 x^{3} - 127 x^{2} - 20 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $21$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[17, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(295273090543420857732075123373712153=7^{14}\cdot 27427\cdot 15873511208841403811\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.87$ | 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: | $7, 27427, 15873511208841403811$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$
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: | $18$ | 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: | \( 42933689957.7 \) (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 384072192000 |
| The 1165 conjugacy class representatives for t21n159 are not computed |
| Character table for t21n159 is not computed |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 42 siblings: | 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ | ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | $18{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $7$ | 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 7.6.4.3 | $x^{6} + 56 x^{3} + 1323$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 7.9.6.1 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 27427 | Data not computed | ||||||
| 15873511208841403811 | Data not computed | ||||||