Normalized defining polynomial
\( x^{24} - 2 x^{22} + 4 x^{20} - 8 x^{18} + 16 x^{16} - 32 x^{14} + 64 x^{12} - 128 x^{10} + 256 x^{8} + \cdots + 4096 \)
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: | \(220715887053399008657112652614467584\) \(\medspace = 2^{36}\cdot 13^{22}\) | 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: | \(29.69\) | 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: | $2^{3/2}13^{11/12}\approx 29.69338662673638$ | ||
Ramified primes: | \(2\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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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: | \(104=2^{3}\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{104}(1,·)$, $\chi_{104}(83,·)$, $\chi_{104}(3,·)$, $\chi_{104}(97,·)$, $\chi_{104}(9,·)$, $\chi_{104}(11,·)$, $\chi_{104}(17,·)$, $\chi_{104}(75,·)$, $\chi_{104}(19,·)$, $\chi_{104}(89,·)$, $\chi_{104}(25,·)$, $\chi_{104}(27,·)$, $\chi_{104}(33,·)$, $\chi_{104}(67,·)$, $\chi_{104}(35,·)$, $\chi_{104}(81,·)$, $\chi_{104}(41,·)$, $\chi_{104}(43,·)$, $\chi_{104}(99,·)$, $\chi_{104}(49,·)$, $\chi_{104}(51,·)$, $\chi_{104}(73,·)$, $\chi_{104}(57,·)$, $\chi_{104}(59,·)$$\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
$C_{9}$, which has order $9$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1}{2048} a^{22} \) (order $26$) | 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: | $\frac{1}{64}a^{12}+\frac{1}{4}a^{4}$, $\frac{1}{2048}a^{22}-\frac{1}{1024}a^{20}+\frac{1}{128}a^{14}-\frac{1}{64}a^{12}+\frac{1}{8}a^{6}-\frac{1}{4}a^{4}+\frac{1}{2}a^{2}$, $\frac{1}{2048}a^{22}+\frac{1}{128}a^{14}-\frac{1}{4}a^{4}$, $\frac{1}{2048}a^{22}-\frac{1}{1024}a^{20}-\frac{1}{16}a^{8}$, $\frac{1}{1024}a^{20}-\frac{1}{128}a^{14}+\frac{1}{64}a^{12}+\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+1$, $\frac{1}{2048}a^{23}-\frac{1}{1024}a^{21}-\frac{1}{256}a^{17}-\frac{1}{128}a^{14}-\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{1}{32}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-a$, $\frac{1}{512}a^{18}+\frac{1}{64}a^{13}+\frac{1}{16}a^{8}$, $\frac{1}{512}a^{18}-\frac{1}{128}a^{14}+\frac{1}{2}a^{3}$, $\frac{1}{256}a^{17}-\frac{1}{16}a^{8}+1$, $\frac{1}{1024}a^{20}-\frac{1}{256}a^{17}+\frac{1}{128}a^{14}$, $\frac{1}{1024}a^{20}+\frac{1}{512}a^{19}+\frac{1}{512}a^{18}$ (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: | \( 60385547.84262381 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 60385547.84262381 \cdot 9}{26\cdot\sqrt{220715887053399008657112652614467584}}\cr\approx \mathstrut & 0.168439442448529 \end{aligned}\] (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 | R | ${\href{/padicField/3.3.0.1}{3} }^{8}$ | ${\href{/padicField/5.4.0.1}{4} }^{6}$ | ${\href{/padicField/7.12.0.1}{12} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{6}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{6}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 $24$ | $2$ | $12$ | $36$ | |||
\(13\) | Deg $24$ | $12$ | $2$ | $22$ |