magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -4, 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, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 4*x + 5)
gp: K = bnfinit(x^42 - 4*x + 5, 1)
Normalized defining polynomial
\( x^{42} - 4 x + 5 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $42$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 21]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-6827138870565194341435326707627876755227739584667606837527183576928421951964201120532126649810944=-\,2^{42}\cdot 257\cdot 3481561\cdot 654827729\cdot 761992291147\cdot 1236911477052699683\cdot 2810965583895622035799170714070354517\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $202.11$ | 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: | $2, 257, 3481561, 654827729, 761992291147, 1236911477052699683, 2810965583895622035799170714070354517$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$
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
$S_{42}$ (as 42T9491):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 1405006117752879898543142606244511569936384000000000 |
| The 53174 conjugacy class representatives for $S_{42}$ are not computed |
| Character table for $S_{42}$ 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 | R | $37{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | $20^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/LocalNumberField/7.14.0.1}{14} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | $39{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ | $17{,}\,{\href{/LocalNumberField/13.13.0.1}{13} }{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | $24{,}\,{\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | $18{,}\,17{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ | $41{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $30{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $17{,}\,{\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | $42$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $23{,}\,16{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | $28{,}\,{\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | $21{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.4.4.2 | $x^{4} - x^{2} + 5$ | $2$ | $2$ | $4$ | $C_4$ | $[2]^{2}$ | |
| 2.6.6.5 | $x^{6} - 2 x^{4} + x^{2} - 3$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 2 x^{4} + x^{2} - 7$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.12.12.26 | $x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
| 2.12.12.25 | $x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ | |
| 257 | Data not computed | ||||||
| 3481561 | Data not computed | ||||||
| 654827729 | Data not computed | ||||||
| 761992291147 | Data not computed | ||||||
| 1236911477052699683 | Data not computed | ||||||
| 2810965583895622035799170714070354517 | Data not computed | ||||||