Normalized defining polynomial
\( x^{24} - 2 x^{23} + 2 x^{22} + 10 x^{21} - 20 x^{20} + 13 x^{19} + 57 x^{18} - 98 x^{17} + 19 x^{16} + 228 x^{15} - 267 x^{14} - 159 x^{13} + 711 x^{12} - 318 x^{11} - 1068 x^{10} + 1824 x^{9} + 304 x^{8} - 3136 x^{7} + 3648 x^{6} + 1664 x^{5} - 5120 x^{4} + 5120 x^{3} + 2048 x^{2} - 4096 x + 4096 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(138359014736314946502328332753681\)\(\medspace = 3^{12}\cdot 7^{20}\cdot 239^{4}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $21.84$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $3, 7, 239$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $8$ | ||
This field is not Galois over $\Q$. | |||
This is a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{1}{4} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{3}{16} a^{11} + \frac{3}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{8} a^{7} + \frac{5}{16} a^{6} - \frac{5}{16} a^{5} - \frac{7}{16} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{32} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{3}{32} a^{12} - \frac{5}{32} a^{11} + \frac{7}{32} a^{9} - \frac{7}{16} a^{8} - \frac{11}{32} a^{7} + \frac{3}{32} a^{6} - \frac{7}{32} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{64} a^{18} - \frac{1}{32} a^{16} - \frac{1}{32} a^{15} + \frac{5}{64} a^{13} + \frac{3}{64} a^{12} - \frac{1}{64} a^{10} + \frac{5}{32} a^{9} - \frac{19}{64} a^{8} - \frac{21}{64} a^{7} + \frac{25}{64} a^{6} - \frac{3}{8} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{128} a^{19} - \frac{1}{64} a^{17} - \frac{1}{64} a^{16} + \frac{5}{128} a^{14} - \frac{13}{128} a^{13} - \frac{1}{8} a^{12} + \frac{31}{128} a^{11} + \frac{13}{64} a^{10} + \frac{29}{128} a^{9} + \frac{59}{128} a^{8} + \frac{9}{128} a^{7} + \frac{1}{4} a^{6} - \frac{3}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{512} a^{20} - \frac{1}{256} a^{19} - \frac{1}{256} a^{18} + \frac{1}{256} a^{17} + \frac{1}{128} a^{16} + \frac{5}{512} a^{15} + \frac{41}{512} a^{14} + \frac{21}{256} a^{13} - \frac{33}{512} a^{12} + \frac{23}{128} a^{11} - \frac{119}{512} a^{10} + \frac{33}{512} a^{9} - \frac{141}{512} a^{8} - \frac{41}{256} a^{7} - \frac{3}{64} a^{6} + \frac{29}{64} a^{5} - \frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1024} a^{21} + \frac{1}{512} a^{19} - \frac{1}{512} a^{18} - \frac{1}{128} a^{17} - \frac{3}{1024} a^{16} + \frac{51}{1024} a^{15} + \frac{9}{256} a^{14} + \frac{75}{1024} a^{13} - \frac{51}{512} a^{12} + \frac{185}{1024} a^{11} - \frac{125}{1024} a^{10} - \frac{227}{1024} a^{9} + \frac{123}{256} a^{8} - \frac{29}{256} a^{7} - \frac{41}{128} a^{6} + \frac{29}{64} a^{5} + \frac{7}{16} a^{4} - \frac{5}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{176128} a^{22} + \frac{19}{44032} a^{21} + \frac{83}{88064} a^{20} + \frac{39}{88064} a^{19} - \frac{165}{22016} a^{18} + \frac{613}{176128} a^{17} - \frac{2465}{176128} a^{16} + \frac{127}{44032} a^{15} + \frac{4927}{176128} a^{14} - \frac{10685}{88064} a^{13} - \frac{11763}{176128} a^{12} - \frac{41249}{176128} a^{11} + \frac{12293}{176128} a^{10} + \frac{8171}{44032} a^{9} - \frac{319}{22016} a^{8} + \frac{343}{1376} a^{7} - \frac{5403}{11008} a^{6} + \frac{2301}{5504} a^{5} + \frac{155}{2752} a^{4} - \frac{575}{1376} a^{3} - \frac{3}{172} a^{2} + \frac{109}{344} a + \frac{45}{172}$, $\frac{1}{44736512} a^{23} + \frac{7}{22368256} a^{22} - \frac{6745}{22368256} a^{21} - \frac{17491}{22368256} a^{20} - \frac{10139}{11184128} a^{19} + \frac{28789}{44736512} a^{18} - \frac{656919}{44736512} a^{17} + \frac{7599}{520192} a^{16} - \frac{2742105}{44736512} a^{15} + \frac{789297}{11184128} a^{14} - \frac{39205}{1040384} a^{13} - \frac{2106599}{44736512} a^{12} + \frac{67475}{44736512} a^{11} - \frac{2905869}{22368256} a^{10} - \frac{305561}{2796032} a^{9} - \frac{1375407}{2796032} a^{8} + \frac{714569}{2796032} a^{7} - \frac{303565}{699008} a^{6} + \frac{15273}{174752} a^{5} + \frac{741}{5461} a^{4} - \frac{42903}{174752} a^{3} + \frac{33849}{87376} a^{2} - \frac{7859}{21844} a + \frac{2905}{21844}$
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( \frac{233629}{22368256} a^{23} + \frac{356283}{22368256} a^{22} - \frac{244523}{11184128} a^{21} + \frac{59507}{699008} a^{20} + \frac{1973597}{11184128} a^{19} - \frac{5059799}{22368256} a^{18} + \frac{2173295}{11184128} a^{17} + \frac{22093205}{22368256} a^{16} - \frac{502931}{520192} a^{15} - \frac{13893057}{22368256} a^{14} + \frac{77191467}{22368256} a^{13} - \frac{16518885}{11184128} a^{12} - \frac{59569811}{11184128} a^{11} + \frac{181994879}{22368256} a^{10} + \frac{23259973}{5592064} a^{9} - \frac{43889855}{2796032} a^{8} + \frac{18372467}{1398016} a^{7} + \frac{35452205}{1398016} a^{6} - \frac{16496667}{699008} a^{5} + \frac{4517191}{349504} a^{4} + \frac{9277671}{174752} a^{3} - \frac{653149}{43688} a^{2} + \frac{263005}{43688} a + \frac{1000995}{21844} \) (order $42$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 9137650.497592166 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_2^3\times A_4$ (as 24T135):
A solvable group of order 96 |
The 32 conjugacy class representatives for $C_2^3\times A_4$ |
Character table for $C_2^3\times A_4$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
$3$ | 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
$7$ | 7.12.10.1 | $x^{12} - 70 x^{6} + 35721$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
7.12.10.1 | $x^{12} - 70 x^{6} + 35721$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
239 | Data not computed |