Normalized defining polynomial
\( x^{24} - 2x^{22} + 8x^{18} - 16x^{16} + 64x^{12} - 256x^{8} + 512x^{6} - 2048x^{2} + 4096 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(2914041287899137980901233132568576\)
\(\medspace = 2^{36}\cdot 3^{12}\cdot 7^{20}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.79\) | 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))
| |
Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(168=2^{3}\cdot 3\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(5,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(13,·)$, $\chi_{168}(17,·)$, $\chi_{168}(149,·)$, $\chi_{168}(89,·)$, $\chi_{168}(25,·)$, $\chi_{168}(157,·)$, $\chi_{168}(101,·)$, $\chi_{168}(97,·)$, $\chi_{168}(37,·)$, $\chi_{168}(145,·)$, $\chi_{168}(41,·)$, $\chi_{168}(125,·)$, $\chi_{168}(109,·)$, $\chi_{168}(29,·)$, $\chi_{168}(113,·)$, $\chi_{168}(53,·)$, $\chi_{168}(73,·)$, $\chi_{168}(121,·)$, $\chi_{168}(61,·)$, $\chi_{168}(85,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{6}$, which has order $6$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: |
\( -\frac{1}{2048} a^{22} \)
(order $42$)
| 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: |
$\frac{1}{256}a^{16}+\frac{1}{2}a^{2}-1$, $\frac{1}{2048}a^{22}+\frac{1}{256}a^{16}+\frac{1}{16}a^{8}+\frac{1}{2}a^{2}$, $\frac{1}{32}a^{10}-1$, $\frac{1}{1024}a^{20}-1$, $\frac{1}{2048}a^{22}+\frac{1}{256}a^{16}+\frac{1}{32}a^{10}+\frac{1}{16}a^{8}+\frac{1}{4}a^{4}+\frac{1}{2}a^{2}-1$, $\frac{1}{128}a^{15}+\frac{1}{16}a^{8}+a$, $\frac{1}{2048}a^{22}-\frac{1}{512}a^{19}+\frac{1}{256}a^{16}$, $\frac{1}{2048}a^{23}+\frac{1}{2048}a^{22}+\frac{1}{32}a^{10}+\frac{1}{16}a^{9}$, $\frac{1}{2048}a^{23}-\frac{1}{1024}a^{21}+\frac{1}{256}a^{17}-\frac{1}{128}a^{15}-\frac{1}{64}a^{13}+\frac{1}{32}a^{11}+\frac{1}{32}a^{10}-\frac{1}{8}a^{7}+\frac{1}{4}a^{5}+\frac{1}{2}a^{2}-a$, $\frac{1}{2048}a^{22}+\frac{1}{1024}a^{21}+\frac{1}{8}a^{7}+\frac{1}{8}a^{6}$, $\frac{1}{2048}a^{23}-\frac{1}{2048}a^{22}-\frac{1}{1024}a^{20}+\frac{1}{256}a^{17}-\frac{1}{128}a^{14}+\frac{1}{16}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{6}+\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-a+1$
| 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: | \( 25576950.522236273 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 25576950.522236273 \cdot 6}{42\cdot\sqrt{2914041287899137980901233132568576}}\cr\approx \mathstrut & 0.256248477211597 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_6$ (as 24T3):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2^2\times C_6$ |
Character table for $C_2^2\times C_6$ 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 | ${\href{/padicField/5.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\)
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ | |
7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |