Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^31 + 5*x - 3)
gp: K = bnfinit(x^31 + 5*x - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 5, 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]);
\( x^{31} + 5 x - 3 \)
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: | \(-10\!\cdots\!191\)\(\medspace = -\,3^{30}\cdot 43\cdot 16293623473\cdot 738487392300001178494609875436847461381\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $125.22$ | 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: | $3, 43, 16293623473, 738487392300001178494609875436847461381$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{3} a^{30} - \frac{1}{3} a^{29} + \frac{1}{3} a^{28} - \frac{1}{3} a^{27} + \frac{1}{3} a^{26} - \frac{1}{3} a^{25} + \frac{1}{3} a^{24} - \frac{1}{3} a^{23} + \frac{1}{3} a^{22} - \frac{1}{3} a^{21} + \frac{1}{3} a^{20} - \frac{1}{3} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
Class group and class number
not computed
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: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | not computed | 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 | $21{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $30{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | $17{,}\,{\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | $22{,}\,{\href{/LocalNumberField/19.9.0.1}{9} }$ | $25{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $22{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | $25{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | $18{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
| 3.12.12.15 | $x^{12} + 12 x^{11} + 87 x^{10} + 57 x^{9} + 81 x^{8} - 36 x^{6} + 54 x^{5} + 54 x^{4} + 54 x^{3} - 81 x - 81$ | $3$ | $4$ | $12$ | 12T173 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
| 3.12.12.11 | $x^{12} + 18 x^{11} - 120 x^{10} - 33 x^{9} - 99 x^{8} - 117 x^{7} + 108 x^{6} + 54 x^{4} - 81 x^{2} - 81 x - 81$ | $3$ | $4$ | $12$ | 12T170 | $[3/2, 3/2, 3/2, 3/2]_{2}^{4}$ | |
| 43 | Data not computed | ||||||
| 16293623473 | Data not computed | ||||||
| 738487392300001178494609875436847461381 | Data not computed | ||||||