Normalized defining polynomial
\( x^{24} - 8x^{20} + 48x^{16} - 240x^{12} + 768x^{8} - 2048x^{4} + 4096 \)
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: | \(117090840873653968654439037792157696\) \(\medspace = 2^{52}\cdot 7^{4}\cdot 101^{8}\) | 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: | \(28.92\) | 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: | $2^{13/6}7^{1/2}101^{1/2}\approx 119.38269102516833$ | ||
Ramified primes: | \(2\), \(7\), \(101\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{8}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{9}$, $\frac{1}{8}a^{10}$, $\frac{1}{16}a^{11}-\frac{1}{4}a^{5}$, $\frac{1}{16}a^{12}$, $\frac{1}{32}a^{13}-\frac{1}{16}a^{10}-\frac{1}{8}a^{7}-\frac{1}{4}a^{4}$, $\frac{1}{32}a^{14}-\frac{1}{2}a^{2}$, $\frac{1}{64}a^{15}-\frac{1}{4}a^{3}$, $\frac{1}{64}a^{16}-\frac{1}{4}a^{4}$, $\frac{1}{128}a^{17}+\frac{1}{8}a^{5}$, $\frac{1}{256}a^{18}-\frac{1}{16}a^{10}+\frac{1}{16}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{512}a^{19}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}-\frac{3}{32}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{1536}a^{20}-\frac{1}{32}a^{12}-\frac{1}{32}a^{8}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{3}$, $\frac{1}{3072}a^{21}-\frac{1}{64}a^{13}-\frac{1}{16}a^{10}+\frac{3}{64}a^{9}-\frac{1}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{3}a$, $\frac{1}{6144}a^{22}-\frac{1}{128}a^{16}+\frac{1}{128}a^{14}+\frac{3}{128}a^{10}+\frac{1}{16}a^{6}-\frac{1}{8}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{12288}a^{23}-\frac{1}{3072}a^{20}-\frac{1}{256}a^{17}-\frac{1}{128}a^{16}+\frac{1}{256}a^{15}-\frac{1}{64}a^{14}-\frac{1}{64}a^{12}+\frac{3}{256}a^{11}-\frac{1}{16}a^{9}-\frac{3}{64}a^{8}-\frac{3}{32}a^{7}+\frac{3}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{12}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{3}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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)
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Torsion generator: | \( -\frac{1}{3072} a^{21} - \frac{1}{64} a^{13} + \frac{5}{64} a^{9} - \frac{1}{8} a^{5} + \frac{2}{3} a \) (order $8$) | 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: | $\frac{1}{6144}a^{22}+\frac{1}{1536}a^{20}+\frac{1}{256}a^{18}-\frac{1}{128}a^{16}-\frac{3}{128}a^{14}+\frac{1}{32}a^{12}+\frac{11}{128}a^{10}-\frac{5}{32}a^{8}-\frac{3}{8}a^{6}+\frac{5}{8}a^{4}+\frac{2}{3}a^{2}-\frac{1}{3}$, $\frac{1}{6144}a^{23}+\frac{1}{1536}a^{21}-\frac{1}{512}a^{19}+\frac{1}{128}a^{15}-\frac{1}{128}a^{11}-\frac{1}{32}a^{9}+\frac{1}{32}a^{7}+\frac{5}{12}a^{3}-\frac{5}{6}a$, $\frac{1}{6144}a^{23}-\frac{1}{1024}a^{22}+\frac{1}{256}a^{18}-\frac{1}{128}a^{15}-\frac{1}{64}a^{14}+\frac{3}{128}a^{11}+\frac{3}{64}a^{10}-\frac{3}{16}a^{7}-\frac{1}{16}a^{6}+\frac{5}{12}a^{3}-1$, $\frac{1}{4096}a^{23}-\frac{1}{6144}a^{22}+\frac{1}{1024}a^{21}+\frac{1}{3072}a^{20}+\frac{1}{256}a^{18}-\frac{1}{256}a^{17}-\frac{1}{256}a^{15}-\frac{3}{128}a^{14}+\frac{1}{64}a^{13}+\frac{1}{64}a^{12}+\frac{9}{256}a^{11}+\frac{13}{128}a^{10}-\frac{3}{64}a^{9}-\frac{5}{64}a^{8}-\frac{5}{32}a^{7}-\frac{1}{4}a^{6}+\frac{1}{16}a^{5}+\frac{3}{8}a^{4}+\frac{1}{2}a^{3}+\frac{7}{12}a^{2}+\frac{1}{2}a-\frac{2}{3}$, $\frac{1}{12288}a^{23}-\frac{1}{1536}a^{21}-\frac{1}{3072}a^{20}-\frac{1}{512}a^{19}+\frac{1}{256}a^{17}+\frac{1}{128}a^{16}+\frac{1}{256}a^{15}+\frac{1}{64}a^{14}-\frac{1}{32}a^{13}-\frac{1}{64}a^{12}-\frac{5}{256}a^{11}-\frac{1}{16}a^{10}+\frac{3}{32}a^{9}+\frac{5}{64}a^{8}+\frac{1}{4}a^{6}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{3}a^{3}-\frac{3}{4}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{7}{12288}a^{23}+\frac{1}{768}a^{22}+\frac{1}{1024}a^{21}-\frac{1}{3072}a^{20}-\frac{1}{128}a^{19}-\frac{3}{256}a^{18}-\frac{1}{256}a^{17}+\frac{1}{128}a^{16}+\frac{11}{256}a^{15}+\frac{3}{64}a^{14}+\frac{1}{64}a^{13}-\frac{1}{64}a^{12}-\frac{43}{256}a^{11}-\frac{3}{16}a^{10}-\frac{3}{64}a^{9}+\frac{5}{64}a^{8}+\frac{19}{32}a^{7}+\frac{9}{16}a^{6}+\frac{1}{16}a^{5}-\frac{1}{4}a^{4}-\frac{7}{6}a^{3}-\frac{5}{12}a^{2}+\frac{1}{2}a-\frac{1}{3}$, $\frac{5}{6144}a^{23}-\frac{1}{6144}a^{22}-\frac{7}{3072}a^{21}+\frac{1}{768}a^{20}-\frac{3}{512}a^{19}-\frac{1}{256}a^{18}+\frac{1}{64}a^{17}-\frac{1}{128}a^{16}+\frac{3}{128}a^{15}+\frac{3}{128}a^{14}-\frac{5}{64}a^{13}-\frac{13}{128}a^{11}-\frac{11}{128}a^{10}+\frac{19}{64}a^{9}-\frac{1}{16}a^{8}+\frac{7}{32}a^{7}+\frac{3}{8}a^{6}-\frac{7}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{6}a^{3}-\frac{2}{3}a^{2}+\frac{7}{6}a+\frac{4}{3}$, $\frac{5}{6144}a^{23}-\frac{1}{6144}a^{22}+\frac{7}{3072}a^{21}-\frac{1}{1536}a^{20}-\frac{3}{512}a^{19}+\frac{1}{256}a^{18}-\frac{1}{64}a^{17}+\frac{1}{128}a^{16}+\frac{3}{128}a^{15}-\frac{1}{128}a^{14}+\frac{5}{64}a^{13}-\frac{1}{32}a^{12}-\frac{13}{128}a^{11}+\frac{5}{128}a^{10}-\frac{19}{64}a^{9}+\frac{5}{32}a^{8}+\frac{7}{32}a^{7}-\frac{1}{4}a^{6}+\frac{7}{8}a^{5}-\frac{5}{8}a^{4}-\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{7}{6}a+\frac{4}{3}$, $\frac{1}{4096}a^{23}-\frac{7}{6144}a^{22}-\frac{1}{1024}a^{21}+\frac{1}{3072}a^{20}+\frac{1}{128}a^{18}+\frac{1}{256}a^{17}-\frac{1}{256}a^{15}-\frac{5}{128}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{12}+\frac{9}{256}a^{11}+\frac{19}{128}a^{10}+\frac{3}{64}a^{9}-\frac{5}{64}a^{8}-\frac{5}{32}a^{7}-\frac{5}{16}a^{6}-\frac{1}{16}a^{5}+\frac{3}{8}a^{4}+\frac{1}{2}a^{3}+\frac{7}{12}a^{2}-\frac{1}{2}a-\frac{5}{3}$, $\frac{5}{12288}a^{23}+\frac{7}{6144}a^{22}-\frac{1}{3072}a^{21}-\frac{3}{1024}a^{20}-\frac{1}{128}a^{18}+\frac{1}{256}a^{17}+\frac{1}{64}a^{16}+\frac{1}{256}a^{15}+\frac{5}{128}a^{14}-\frac{3}{64}a^{13}-\frac{5}{64}a^{12}-\frac{1}{256}a^{11}-\frac{19}{128}a^{10}+\frac{9}{64}a^{9}+\frac{21}{64}a^{8}-\frac{3}{32}a^{7}+\frac{5}{16}a^{6}-\frac{9}{16}a^{5}-\frac{5}{8}a^{4}+\frac{1}{6}a^{3}-\frac{13}{12}a^{2}+\frac{7}{6}a+1$, $\frac{1}{12288}a^{23}+\frac{1}{3072}a^{21}-\frac{1}{3072}a^{20}+\frac{1}{512}a^{19}-\frac{1}{256}a^{17}+\frac{1}{128}a^{16}-\frac{3}{256}a^{15}-\frac{1}{64}a^{14}+\frac{1}{64}a^{13}-\frac{1}{64}a^{12}+\frac{11}{256}a^{11}+\frac{1}{16}a^{10}-\frac{1}{64}a^{9}+\frac{5}{64}a^{8}-\frac{3}{16}a^{7}-\frac{1}{4}a^{6}+\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{1}{12}a^{3}+\frac{3}{4}a^{2}+\frac{1}{3}a-\frac{1}{3}$ (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: | \( 139979456.91557625 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 139979456.91557625 \cdot 6}{8\cdot\sqrt{117090840873653968654439037792157696}}\cr\approx \mathstrut & 1.16150885610799 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3\times S_4$ (as 24T400):
A solvable group of order 192 |
The 40 conjugacy class representatives for $C_2^3\times S_4$ |
Character table for $C_2^3\times S_4$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | ${\href{/padicField/5.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{4}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.12.26.64 | $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$ | $12$ | $1$ | $26$ | $S_3 \times C_2^2$ | $[2, 3]_{3}^{2}$ |
2.12.26.64 | $x^{12} + 2 x^{10} + 4 x^{9} + 2 x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$ | $12$ | $1$ | $26$ | $S_3 \times C_2^2$ | $[2, 3]_{3}^{2}$ | |
\(7\) | 7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(101\) | 101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |