magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -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, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 3*x + 5)
gp: K = bnfinit(x^26 - 3*x + 5, 1)
Normalized defining polynomial
\( x^{26} - 3 x + 5 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $26$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 13]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-73386655124093395346715550870539255750179290771484375=-\,5^{23}\cdot 61\cdot 331\cdot 449\cdot 63806909\cdot 3529639661\cdot 3015121007861\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $107.97$ | 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: | $5, 61, 331, 449, 63806909, 3529639661, 3015121007861$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{5} a^{25} - \frac{2}{5} a^{24} - \frac{1}{5} a^{23} + \frac{2}{5} a^{22} + \frac{1}{5} a^{21} - \frac{2}{5} a^{20} - \frac{1}{5} a^{19} + \frac{2}{5} a^{18} + \frac{1}{5} a^{17} - \frac{2}{5} a^{16} - \frac{1}{5} a^{15} + \frac{2}{5} a^{14} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 16503473635912132 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{26}$ (as 26T96):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 403291461126605635584000000 |
| The 2436 conjugacy class representatives for $S_{26}$ are not computed |
| Character table for $S_{26}$ 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.9.0.1}{9} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ | ${\href{/LocalNumberField/3.3.0.1}{3} }^{8}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | R | $21{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ | $26$ | $24{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | $22{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/LocalNumberField/19.8.0.1}{8} }$ | $26$ | ${\href{/LocalNumberField/29.13.0.1}{13} }{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.13.0.1}{13} }{,}\,{\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | $21{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $16{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | $17{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ | ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $23{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $61$ | $\Q_{61}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.3.0.1 | $x^{3} - x + 10$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 61.6.0.1 | $x^{6} - 4 x + 10$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 61.11.0.1 | $x^{11} - x + 10$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
| 331 | Data not computed | ||||||
| 449 | Data not computed | ||||||
| 63806909 | Data not computed | ||||||
| 3529639661 | Data not computed | ||||||
| 3015121007861 | Data not computed | ||||||