Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 2)
gp: K = bnfinit(x^28 + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![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, 0, 1]);
\( x^{28} + 2 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $28$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(4448716805632354662386823820928005747801019383808\)\(\medspace = 2^{83}\cdot 7^{28}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $54.63$ | 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: | $2, 7$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $2$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$
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: | $13$ | 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: | \( 26773418571562.227 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$D_4\times F_7$ (as 28T41):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A solvable group of order 336 |
| The 35 conjugacy class representatives for $D_4\times F_7$ |
| Character table for $D_4\times F_7$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.2048.1, 7.1.52706752.1, 14.0.91029559914971267072.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 28 siblings: | data not computed |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{7}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | $28$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $7$ | 7.14.14.21 | $x^{14} + 28 x^{12} + 42 x^{11} + 42 x^{9} + 21 x^{8} + 29 x^{7} + 21 x^{6} + 35 x^{5} + 7 x^{4} + 14 x^{3} + 28 x^{2} + 42 x + 45$ | $7$ | $2$ | $14$ | $F_7 \times C_2$ | $[7/6]_{6}^{2}$ |
| 7.14.14.21 | $x^{14} + 28 x^{12} + 42 x^{11} + 42 x^{9} + 21 x^{8} + 29 x^{7} + 21 x^{6} + 35 x^{5} + 7 x^{4} + 14 x^{3} + 28 x^{2} + 42 x + 45$ | $7$ | $2$ | $14$ | $F_7 \times C_2$ | $[7/6]_{6}^{2}$ | |