magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -2, 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^28 - 2*x + 5)
gp: K = bnfinit(x^28 - 2*x + 5, 1)
Normalized defining polynomial
\( x^{28} - 2 x + 5 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 14]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(61738347845594210065221391307477679628500729234048268894208=2^{26}\cdot 53\cdot 137\cdot 5167\cdot 182453\cdot 1981277\cdot 67833466448141248837857216309551\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $125.79$ | 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: | $2, 53, 137, 5167, 182453, 1981277, 67833466448141248837857216309551$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{26} - \frac{1}{2} a^{25} - \frac{1}{2} a^{24} - \frac{1}{2} a^{23} - \frac{1}{2} a^{22} - \frac{1}{2} a^{21} - \frac{1}{2} a^{20} - \frac{1}{2} a^{19} - \frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$
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: | $13$ | 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: | \( 7257941571662790000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{28}$ (as 28T1854):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 304888344611713860501504000000 |
| The 3718 conjugacy class representatives for $S_{28}$ are not computed |
| Character table for $S_{28}$ 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 | ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | $18{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | $27{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | $19{,}\,{\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $19{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $23{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $15{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ | $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | R | $17{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.12.12.32 | $x^{12} + 6 x^{11} - 4 x^{10} - 2 x^{9} + 4 x^{8} - 8 x^{7} - 8 x^{6} - 8 x^{5} - 4 x^{4} - 8$ | $4$ | $3$ | $12$ | 12T129 | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
| 2.12.12.32 | $x^{12} + 6 x^{11} - 4 x^{10} - 2 x^{9} + 4 x^{8} - 8 x^{7} - 8 x^{6} - 8 x^{5} - 4 x^{4} - 8$ | $4$ | $3$ | $12$ | 12T129 | $[4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
| 53 | Data not computed | ||||||
| $137$ | $\Q_{137}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 137.2.1.1 | $x^{2} - 137$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.6.0.1 | $x^{6} - x + 29$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 137.8.0.1 | $x^{8} - x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 137.11.0.1 | $x^{11} - x + 26$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
| 5167 | Data not computed | ||||||
| 182453 | Data not computed | ||||||
| 1981277 | Data not computed | ||||||
| 67833466448141248837857216309551 | Data not computed | ||||||