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)
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Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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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)
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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))
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Galois root discriminant: | $2^{3/2}3^{1/2}7^{5/6}\approx 24.794421938893013$ | ||
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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(168=2^{3}\cdot 3\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(131,·)$, $\chi_{168}(65,·)$, $\chi_{168}(137,·)$, $\chi_{168}(139,·)$, $\chi_{168}(11,·)$, $\chi_{168}(145,·)$, $\chi_{168}(19,·)$, $\chi_{168}(115,·)$, $\chi_{168}(89,·)$, $\chi_{168}(25,·)$, $\chi_{168}(155,·)$, $\chi_{168}(107,·)$, $\chi_{168}(97,·)$, $\chi_{168}(67,·)$, $\chi_{168}(163,·)$, $\chi_{168}(17,·)$, $\chi_{168}(41,·)$, $\chi_{168}(43,·)$, $\chi_{168}(113,·)$, $\chi_{168}(83,·)$, $\chi_{168}(73,·)$, $\chi_{168}(121,·)$, $\chi_{168}(59,·)$$\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
$C_{7}$, which has order $7$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{2048} a^{22} + \frac{1}{1024} a^{20} - \frac{1}{256} a^{16} - \frac{1}{128} a^{14} + \frac{1}{64} a^{12} + \frac{1}{32} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{4} + 1 \) (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)
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Fundamental units: | $\frac{1}{64}a^{12}-\frac{1}{8}a^{6}$, $\frac{1}{2048}a^{22}+\frac{1}{32}a^{10}$, $\frac{1}{2048}a^{22}-\frac{1}{1024}a^{20}-\frac{1}{16}a^{8}$, $\frac{1}{2048}a^{22}+\frac{1}{1024}a^{20}+\frac{1}{512}a^{18}-\frac{1}{256}a^{16}-\frac{1}{128}a^{14}+\frac{1}{32}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{4}+1$, $\frac{1}{1024}a^{20}+\frac{1}{512}a^{18}+\frac{1}{256}a^{16}+\frac{1}{16}a^{8}-\frac{1}{2}a^{2}-1$, $\frac{1}{1024}a^{20}-\frac{1}{512}a^{18}-\frac{1}{256}a^{16}+\frac{1}{128}a^{15}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-a+1$, $\frac{1}{512}a^{18}+\frac{1}{256}a^{17}-\frac{1}{4}a^{4}+\frac{1}{2}a^{2}$, $\frac{1}{2048}a^{22}+\frac{1}{1024}a^{20}-\frac{1}{512}a^{18}-\frac{1}{256}a^{16}+\frac{1}{128}a^{14}+\frac{1}{64}a^{12}-\frac{1}{16}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}+\frac{1}{2}a^{2}+a$, $\frac{1}{1024}a^{20}-\frac{1}{128}a^{15}+\frac{1}{32}a^{10}$, $\frac{1}{128}a^{15}+\frac{1}{64}a^{12}-\frac{1}{4}a^{4}-a$, $\frac{1}{2048}a^{23}+\frac{1}{2048}a^{22}-\frac{1}{1024}a^{20}-\frac{1}{512}a^{19}+\frac{1}{256}a^{16}-\frac{1}{16}a^{9}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}+a$ (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)]
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Regulator: | \( 19752911.509995345 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 19752911.509995345 \cdot 7}{42\cdot\sqrt{2914041287899137980901233132568576}}\cr\approx \mathstrut & 0.230882192371594 \end{aligned}\] (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 $C_2^2\times C_6$ |
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 | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{12}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.1.0.1}{1} }^{24}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{4}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
\(3\) | 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 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(7\) | 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |