Normalized defining polynomial
\( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \)
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: | \(47330370277129322496000000000000000000000000\) \(\medspace = 2^{64}\cdot 3^{16}\cdot 5^{24}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.17\) | 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: | $2^{2}3^{1/2}5^{3/4}\approx 23.165843705765383$ | ||
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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(120=2^{3}\cdot 3\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{120}(1,·)$, $\chi_{120}(7,·)$, $\chi_{120}(11,·)$, $\chi_{120}(13,·)$, $\chi_{120}(17,·)$, $\chi_{120}(19,·)$, $\chi_{120}(23,·)$, $\chi_{120}(29,·)$, $\chi_{120}(31,·)$, $\chi_{120}(37,·)$, $\chi_{120}(41,·)$, $\chi_{120}(43,·)$, $\chi_{120}(47,·)$, $\chi_{120}(49,·)$, $\chi_{120}(53,·)$, $\chi_{120}(59,·)$, $\chi_{120}(61,·)$, $\chi_{120}(67,·)$, $\chi_{120}(71,·)$, $\chi_{120}(73,·)$, $\chi_{120}(77,·)$, $\chi_{120}(79,·)$, $\chi_{120}(83,·)$, $\chi_{120}(89,·)$, $\chi_{120}(91,·)$, $\chi_{120}(97,·)$, $\chi_{120}(101,·)$, $\chi_{120}(103,·)$, $\chi_{120}(107,·)$, $\chi_{120}(109,·)$, $\chi_{120}(113,·)$, $\chi_{120}(119,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( a \) (order $120$) | 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: | $a^{24}+1$, $a^{16}-1$, $a^{29}-a^{25}-a^{21}-a^{17}+a^{5}+a$, $a^{18}-1$, $a^{6}-1$, $a^{30}-a^{24}-a^{14}+a^{12}-a^{10}+a^{2}$, $a^{31}-a^{27}-a^{25}+a^{9}+a^{7}$, $a^{9}-1$, $a^{9}-a^{6}$, $a^{21}-1$, $a^{29}-a^{28}-a^{24}+a^{16}+a^{12}+a^{8}-1$, $a^{13}-1$, $a^{11}-1$, $a^{31}+a^{30}-a^{10}$, $a^{25}-1$ (assuming GRH) | 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: | \( 5926511257.21094 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 5926511257.21094 \cdot 4}{120\cdot\sqrt{47330370277129322496000000000000000000000000}}\cr\approx \mathstrut & 0.169428240847859 \end{aligned}\] (assuming GRH)
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 $C_2^3\times C_4$ |
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 | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.2.0.1}{2} }^{16}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.2.0.1}{2} }^{16}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{16}$ |
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 $16$ | $4$ | $4$ | $32$ | |||
Deg $16$ | $4$ | $4$ | $32$ | ||||
\(3\) | 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(5\) | 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |