Normalized defining polynomial
\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \)
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)
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Discriminant: | \(572565594852444156646728515625\) \(\medspace = 3^{36}\cdot 5^{18}\) | 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: | \(17.37\) | 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: | $3^{3/2}5^{3/4}\approx 17.374382779324037$ | ||
Ramified primes: | \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(45=3^{2}\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{45}(1,·)$, $\chi_{45}(2,·)$, $\chi_{45}(4,·)$, $\chi_{45}(7,·)$, $\chi_{45}(8,·)$, $\chi_{45}(11,·)$, $\chi_{45}(13,·)$, $\chi_{45}(14,·)$, $\chi_{45}(16,·)$, $\chi_{45}(17,·)$, $\chi_{45}(19,·)$, $\chi_{45}(22,·)$, $\chi_{45}(23,·)$, $\chi_{45}(26,·)$, $\chi_{45}(28,·)$, $\chi_{45}(29,·)$, $\chi_{45}(31,·)$, $\chi_{45}(32,·)$, $\chi_{45}(34,·)$, $\chi_{45}(37,·)$, $\chi_{45}(38,·)$, $\chi_{45}(41,·)$, $\chi_{45}(43,·)$, $\chi_{45}(44,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a \) (order $90$) | 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: | $a^{9}+1$, $a^{6}-1$, $a^{12}-1$, $a^{22}-a^{17}-a^{2}$, $a^{20}-a^{15}+a^{5}$, $a^{2}-1$, $a-1$, $a^{22}+a^{20}-a^{16}+a^{13}+a^{12}-a^{10}+a^{7}+a^{4}-a$, $a^{4}-1$, $a^{21}-a^{18}-a^{14}-a^{9}+a^{6}-1$, $a^{22}-a^{18}-a^{16}+a^{13}+a^{4}-a$ (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)]
| |
Regulator: | \( 2124832.236129185 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 2124832.236129185 \cdot 1}{90\cdot\sqrt{572565594852444156646728515625}}\cr\approx \mathstrut & 0.118121222580884 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{12}$ (as 24T2):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2\times C_{12}$ |
Character table for $C_2\times C_{12}$ |
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 | ${\href{/padicField/2.12.0.1}{12} }^{2}$ | R | R | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{6}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $24$ | $6$ | $4$ | $36$ | |||
\(5\) | Deg $24$ | $4$ | $6$ | $18$ |