Normalized defining polynomial
\( x^{30} - x^{29} - 30 x^{28} + 30 x^{27} + 404 x^{26} - 404 x^{25} - 3223 x^{24} + 3223 x^{23} + 16927 x^{22} - 16927 x^{21} - 61503 x^{20} + 61503 x^{19} + 158101 x^{18} - 158101 x^{17} - 288950 x^{16} + 288950 x^{15} + 371908 x^{14} - 371908 x^{13} - 329002 x^{12} + 329002 x^{11} + 191674 x^{10} - 191674 x^{9} - 68664 x^{8} + 68664 x^{7} + 13548 x^{6} - 13548 x^{5} - 1208 x^{4} + 1208 x^{3} + 32 x^{2} - 32 x + 1 \)
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[30, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(254863675513583304379979153001217903140700917991797=3^{15}\cdot 31^{29}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $47.89$ | 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: | $3, 31$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(93=3\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{93}(64,·)$, $\chi_{93}(1,·)$, $\chi_{93}(67,·)$, $\chi_{93}(4,·)$, $\chi_{93}(68,·)$, $\chi_{93}(7,·)$, $\chi_{93}(10,·)$, $\chi_{93}(11,·)$, $\chi_{93}(76,·)$, $\chi_{93}(77,·)$, $\chi_{93}(16,·)$, $\chi_{93}(17,·)$, $\chi_{93}(82,·)$, $\chi_{93}(19,·)$, $\chi_{93}(86,·)$, $\chi_{93}(23,·)$, $\chi_{93}(25,·)$, $\chi_{93}(26,·)$, $\chi_{93}(28,·)$, $\chi_{93}(29,·)$, $\chi_{93}(70,·)$, $\chi_{93}(65,·)$, $\chi_{93}(40,·)$, $\chi_{93}(92,·)$, $\chi_{93}(44,·)$, $\chi_{93}(49,·)$, $\chi_{93}(83,·)$, $\chi_{93}(53,·)$, $\chi_{93}(89,·)$, $\chi_{93}(74,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $29$ | 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: | \( 3165460564789336.0 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{93}) \), 3.3.961.1, 5.5.923521.1, 6.6.772987077.1, 10.10.6424828185043053.1, \(\Q(\zeta_{31})^+\) |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{3}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{5}$ | $15^{2}$ | $15^{2}$ | $30$ | $15^{2}$ | $15^{2}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }^{5}$ | $30$ | $30$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ | $15^{2}$ | $30$ |
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 | ||||||
| 31 | Data not computed | ||||||