Normalized defining polynomial
\( x^{32} - 175 x^{24} + 30000 x^{16} - 109375 x^{8} + 390625 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(203282392447840896882957090816000000000000000000000000=2^{96}\cdot 3^{16}\cdot 5^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $46.33$ | 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: | $2, 3, 5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(240=2^{4}\cdot 3\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(133,·)$, $\chi_{240}(13,·)$, $\chi_{240}(151,·)$, $\chi_{240}(157,·)$, $\chi_{240}(31,·)$, $\chi_{240}(161,·)$, $\chi_{240}(163,·)$, $\chi_{240}(37,·)$, $\chi_{240}(41,·)$, $\chi_{240}(43,·)$, $\chi_{240}(173,·)$, $\chi_{240}(49,·)$, $\chi_{240}(53,·)$, $\chi_{240}(187,·)$, $\chi_{240}(191,·)$, $\chi_{240}(67,·)$, $\chi_{240}(197,·)$, $\chi_{240}(71,·)$, $\chi_{240}(203,·)$, $\chi_{240}(77,·)$, $\chi_{240}(79,·)$, $\chi_{240}(209,·)$, $\chi_{240}(83,·)$, $\chi_{240}(89,·)$, $\chi_{240}(199,·)$, $\chi_{240}(227,·)$, $\chi_{240}(107,·)$, $\chi_{240}(239,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$, $\chi_{240}(119,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{1875} a^{16} + \frac{1}{75} a^{8} + \frac{1}{3}$, $\frac{1}{1875} a^{17} + \frac{1}{75} a^{9} + \frac{1}{3} a$, $\frac{1}{9375} a^{18} + \frac{4}{375} a^{10} + \frac{1}{15} a^{2}$, $\frac{1}{9375} a^{19} + \frac{4}{375} a^{11} + \frac{1}{15} a^{3}$, $\frac{1}{9375} a^{20} + \frac{1}{375} a^{12} + \frac{1}{15} a^{4}$, $\frac{1}{9375} a^{21} + \frac{1}{375} a^{13} + \frac{1}{15} a^{5}$, $\frac{1}{46875} a^{22} + \frac{4}{1875} a^{14} + \frac{1}{75} a^{6}$, $\frac{1}{46875} a^{23} + \frac{4}{1875} a^{15} + \frac{1}{75} a^{7}$, $\frac{1}{2250000} a^{24} - \frac{55}{144}$, $\frac{1}{2250000} a^{25} - \frac{55}{144} a$, $\frac{1}{11250000} a^{26} - \frac{199}{720} a^{2}$, $\frac{1}{11250000} a^{27} - \frac{199}{720} a^{3}$, $\frac{1}{11250000} a^{28} - \frac{11}{144} a^{4}$, $\frac{1}{11250000} a^{29} - \frac{11}{144} a^{5}$, $\frac{1}{56250000} a^{30} - \frac{199}{3600} a^{6}$, $\frac{1}{56250000} a^{31} - \frac{199}{3600} a^{7}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{203}{56250000} a^{30} + \frac{29}{46875} a^{22} - \frac{199}{1875} a^{14} + \frac{29}{3600} a^{6} \) (order $24$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3\times C_4$ (as 32T34):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^3\times C_4$ |
| Character table for $C_2^3\times C_4$ is not computed |
Intermediate fields
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 | R | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |