Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 - x - 4)
gp: K = bnfinit(x^31 - x - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 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]);
\( x^{31} - x - 4 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $31$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[1, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-18327886165296381817189229292602149543677096810012151519\)\(\medspace = -\,23\cdot 1693\cdot 3697\cdot 19843\cdot 14132537\cdot 75771043\cdot 5991663366536301415229073661\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $60.63$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $23, 1693, 3697, 19843, 14132537, 75771043, 5991663366536301415229073661$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $1$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{14}$, $\frac{1}{2} a^{30} - \frac{1}{2} a^{15}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 4169052629874545.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$S_{31}$ (as 31T12):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A non-solvable group of order 8222838654177922817725562880000000 |
| The 6842 conjugacy class representatives for $S_{31}$ are not computed |
| Character table for $S_{31}$ 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | $29{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | $24{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | $23{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | $30{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $28{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | R | $28{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ | $31$ | $30{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | $26{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ | $25{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ | $30{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 23 | Data not computed | ||||||
| 1693 | Data not computed | ||||||
| 3697 | Data not computed | ||||||
| 19843 | Data not computed | ||||||
| 14132537 | Data not computed | ||||||
| 75771043 | Data not computed | ||||||
| 5991663366536301415229073661 | Data not computed | ||||||