Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 5)
gp: K = bnfinit(x^33 - 5, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-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, 0, 0, 0, 1]);
\(x^{33} - 5\) 
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[1, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(300\!\cdots\!625\)\(\medspace = 3^{33}\cdot 5^{32}\cdot 11^{33}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $157.15$ | 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: | $3, 5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$
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: | $16$ | 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
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A solvable group of order 660 |
| The 33 conjugacy class representatives for t33n11 |
| Character table for t33n11 is not computed |
Intermediate fields
| 3.1.675.1, 11.1.2786246783310546875.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | R | R | $30{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ | $30{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ | $22{,}\,{\href{/LocalNumberField/23.11.0.1}{11} }$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $15^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ | $15^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
| 3.15.15.49 | $x^{15} + 15 x^{14} + 24 x^{13} + 6 x^{12} + 21 x^{11} + 3 x^{10} + x^{9} + 3 x^{8} + 15 x^{7} + 9 x^{6} + 18 x^{5} + 25 x^{3} + 21 x^{2} + 9 x + 4$ | $3$ | $5$ | $15$ | $S_3 \times C_5$ | $[3/2]_{2}^{5}$ | |
| 3.15.15.49 | $x^{15} + 15 x^{14} + 24 x^{13} + 6 x^{12} + 21 x^{11} + 3 x^{10} + x^{9} + 3 x^{8} + 15 x^{7} + 9 x^{6} + 18 x^{5} + 25 x^{3} + 21 x^{2} + 9 x + 4$ | $3$ | $5$ | $15$ | $S_3 \times C_5$ | $[3/2]_{2}^{5}$ | |
| 5 | Data not computed | ||||||
| 11 | Data not computed | ||||||