Normalized defining polynomial
\( x^{28} - x^{27} - 28 x^{26} + 28 x^{25} + 349 x^{24} - 349 x^{23} - 2551 x^{22} + 2551 x^{21} + 12123 x^{20} - 12123 x^{19} - 39236 x^{18} + 39236 x^{17} + 88045 x^{16} - 88045 x^{15} - 136763 x^{14} + 136763 x^{13} + 144247 x^{12} - 144247 x^{11} - 99295 x^{10} + 99295 x^{9} + 41703 x^{8} - 41703 x^{7} - 9569 x^{6} + 9569 x^{5} + 987 x^{4} - 987 x^{3} - 28 x^{2} + 28 x + 1 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[28, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(14603047886206093768209337615200705673567789821=3^{14}\cdot 29^{27}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $44.54$ | 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: | $3, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(87=3\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{87}(64,·)$, $\chi_{87}(1,·)$, $\chi_{87}(2,·)$, $\chi_{87}(67,·)$, $\chi_{87}(4,·)$, $\chi_{87}(68,·)$, $\chi_{87}(7,·)$, $\chi_{87}(8,·)$, $\chi_{87}(11,·)$, $\chi_{87}(13,·)$, $\chi_{87}(14,·)$, $\chi_{87}(77,·)$, $\chi_{87}(16,·)$, $\chi_{87}(17,·)$, $\chi_{87}(82,·)$, $\chi_{87}(22,·)$, $\chi_{87}(25,·)$, $\chi_{87}(26,·)$, $\chi_{87}(28,·)$, $\chi_{87}(32,·)$, $\chi_{87}(34,·)$, $\chi_{87}(41,·)$, $\chi_{87}(44,·)$, $\chi_{87}(47,·)$, $\chi_{87}(49,·)$, $\chi_{87}(50,·)$, $\chi_{87}(52,·)$, $\chi_{87}(56,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $27$ | 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: | \( 114482618238342.14 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{29}) \), 4.4.219501.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\) |
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 | $28$ | R | ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ | $28$ | ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ | $28$ | ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ | R | $28$ | $28$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ | $28$ | $28$ | ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 29 | Data not computed | ||||||