Normalized defining polynomial
\( x^{32} - 47 x^{24} + 2208 x^{16} - 47 x^{8} + 1 \)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(520402924666472696020370152488960000000000000000\)\(\medspace = 2^{96}\cdot 3^{16}\cdot 5^{16}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $30.98$ | 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: | $2, 3, 5$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $32$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(240=2^{4}\cdot 3\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(131,·)$, $\chi_{240}(11,·)$, $\chi_{240}(19,·)$, $\chi_{240}(149,·)$, $\chi_{240}(151,·)$, $\chi_{240}(29,·)$, $\chi_{240}(31,·)$, $\chi_{240}(161,·)$, $\chi_{240}(41,·)$, $\chi_{240}(71,·)$, $\chi_{240}(49,·)$, $\chi_{240}(179,·)$, $\chi_{240}(181,·)$, $\chi_{240}(59,·)$, $\chi_{240}(61,·)$, $\chi_{240}(191,·)$, $\chi_{240}(139,·)$, $\chi_{240}(199,·)$, $\chi_{240}(119,·)$, $\chi_{240}(79,·)$, $\chi_{240}(209,·)$, $\chi_{240}(211,·)$, $\chi_{240}(89,·)$, $\chi_{240}(91,·)$, $\chi_{240}(221,·)$, $\chi_{240}(101,·)$, $\chi_{240}(229,·)$, $\chi_{240}(109,·)$, $\chi_{240}(239,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{21} a^{16} + \frac{8}{21} a^{8} + \frac{1}{21}$, $\frac{1}{21} a^{17} + \frac{8}{21} a^{9} + \frac{1}{21} a$, $\frac{1}{21} a^{18} + \frac{8}{21} a^{10} + \frac{1}{21} a^{2}$, $\frac{1}{21} a^{19} + \frac{8}{21} a^{11} + \frac{1}{21} a^{3}$, $\frac{1}{21} a^{20} + \frac{8}{21} a^{12} + \frac{1}{21} a^{4}$, $\frac{1}{21} a^{21} + \frac{8}{21} a^{13} + \frac{1}{21} a^{5}$, $\frac{1}{21} a^{22} + \frac{8}{21} a^{14} + \frac{1}{21} a^{6}$, $\frac{1}{21} a^{23} + \frac{8}{21} a^{15} + \frac{1}{21} a^{7}$, $\frac{1}{46368} a^{24} - \frac{17711}{46368}$, $\frac{1}{46368} a^{25} - \frac{17711}{46368} a$, $\frac{1}{46368} a^{26} - \frac{17711}{46368} a^{2}$, $\frac{1}{46368} a^{27} - \frac{17711}{46368} a^{3}$, $\frac{1}{46368} a^{28} - \frac{17711}{46368} a^{4}$, $\frac{1}{46368} a^{29} - \frac{17711}{46368} a^{5}$, $\frac{1}{46368} a^{30} - \frac{17711}{46368} a^{6}$, $\frac{1}{46368} a^{31} - \frac{17711}{46368} a^{7}$
Class group and class number
$C_{20}$, which has order $20$ (assuming GRH)
Unit group
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{4183}{46368} a^{27} + \frac{89}{21} a^{19} - \frac{4181}{21} a^{11} + \frac{89}{46368} a^{3} \) (order $48$) | 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: | \( 82239790500.5115 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$ |
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 | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |