magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^39 - 3*x - 1)
gp: K = bnfinit(x^39 - 3*x - 1, 1)
Normalized defining polynomial
\( x^{39} - 3 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $39$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[3, 18]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4360191917307956891935658158444803764507661447188598787724416645863129369703209=3^{40}\cdot 113\cdot 28201\cdot 37423\cdot 9699563\cdot 158105977\cdot 1960981103314503221804260659585541\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $103.85$ | 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: | $3, 113, 28201, 37423, 9699563, 158105977, 1960981103314503221804260659585541$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Not computed
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $20$ | 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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{39}$ (as 39T306):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 20397882081197443358640281739902897356800000000 |
| The 31185 conjugacy class representatives for $S_{39}$ are not computed |
| Character table for $S_{39}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $26{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ | R | $27{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ | ${\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.13.0.1}{13} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | $16^{2}{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ | $25{,}\,{\href{/LocalNumberField/17.13.0.1}{13} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $29{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | $19{,}\,{\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $28{,}\,{\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $21{,}\,16{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ | $29{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ | $39$ | $33{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/LocalNumberField/53.11.0.1}{11} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.4.1 | $x^{3} - 3 x^{2} + 21$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| 3.9.9.5 | $x^{9} + 3 x^{7} + 3 x^{6} + 54$ | $3$ | $3$ | $9$ | $(C_3^2:C_3):C_2$ | $[3/2, 3/2, 3/2]_{2}^{3}$ | |
| 3.9.9.4 | $x^{9} + 3 x^{6} + 9 x^{4} + 54$ | $3$ | $3$ | $9$ | $(C_3^2:C_3):C_2$ | $[3/2, 3/2, 3/2]_{2}^{3}$ | |
| 3.9.9.7 | $x^{9} + 18 x^{3} + 54 x + 27$ | $3$ | $3$ | $9$ | $C_3^2 : S_3 $ | $[3/2, 3/2]_{2}^{3}$ | |
| 3.9.9.6 | $x^{9} + 3 x^{7} + 3 x^{6} + 18 x^{4} + 54$ | $3$ | $3$ | $9$ | $S_3\times C_3$ | $[3/2]_{2}^{3}$ | |
| 113 | Data not computed | ||||||
| 28201 | Data not computed | ||||||
| 37423 | Data not computed | ||||||
| 9699563 | Data not computed | ||||||
| 158105977 | Data not computed | ||||||
| 1960981103314503221804260659585541 | Data not computed | ||||||