Normalized defining polynomial
\( x^{32} - 175x^{24} + 30000x^{16} - 109375x^{8} + 390625 \)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(203282392447840896882957090816000000000000000000000000\)
\(\medspace = 2^{96}\cdot 3^{16}\cdot 5^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(46.33\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{3}3^{1/2}5^{3/4}\approx 46.331687411530766$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| 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. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{1875}a^{16}+\frac{1}{75}a^{8}+\frac{1}{3}$, $\frac{1}{1875}a^{17}+\frac{1}{75}a^{9}+\frac{1}{3}a$, $\frac{1}{9375}a^{18}+\frac{4}{375}a^{10}+\frac{1}{15}a^{2}$, $\frac{1}{9375}a^{19}+\frac{4}{375}a^{11}+\frac{1}{15}a^{3}$, $\frac{1}{9375}a^{20}+\frac{1}{375}a^{12}+\frac{1}{15}a^{4}$, $\frac{1}{9375}a^{21}+\frac{1}{375}a^{13}+\frac{1}{15}a^{5}$, $\frac{1}{46875}a^{22}+\frac{4}{1875}a^{14}+\frac{1}{75}a^{6}$, $\frac{1}{46875}a^{23}+\frac{4}{1875}a^{15}+\frac{1}{75}a^{7}$, $\frac{1}{2250000}a^{24}-\frac{55}{144}$, $\frac{1}{2250000}a^{25}-\frac{55}{144}a$, $\frac{1}{11250000}a^{26}-\frac{199}{720}a^{2}$, $\frac{1}{11250000}a^{27}-\frac{199}{720}a^{3}$, $\frac{1}{11250000}a^{28}-\frac{11}{144}a^{4}$, $\frac{1}{11250000}a^{29}-\frac{11}{144}a^{5}$, $\frac{1}{56250000}a^{30}-\frac{199}{3600}a^{6}$, $\frac{1}{56250000}a^{31}-\frac{199}{3600}a^{7}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
not computed
Unit group
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{203}{56250000} a^{30} + \frac{29}{46875} a^{22} - \frac{199}{1875} a^{14} + \frac{29}{3600} a^{6} \)
(order $24$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
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 | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{16}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.2.0.1}{2} }^{16}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.2.0.1}{2} }^{16}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{16}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $8$ | $2$ | $48$ | |||
| Deg $16$ | $8$ | $2$ | $48$ | ||||
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(5\)
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.2 | $x^{8} + 10 x^{4} - 25$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |