Normalized defining polynomial
\( x^{32} - x^{28} - 15 x^{24} + 31 x^{20} + 209 x^{16} + 496 x^{12} - 3840 x^{8} - 4096 x^{4} + 65536 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(36539993586348786372837376000000000000000000000000\)\(\medspace = 2^{64}\cdot 5^{24}\cdot 7^{16}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $35.39$ | 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: | $2, 5, 7$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Gal(K/\Q)|$: | $32$ | ||
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(280=2^{3}\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(139,·)$, $\chi_{280}(13,·)$, $\chi_{280}(279,·)$, $\chi_{280}(153,·)$, $\chi_{280}(27,·)$, $\chi_{280}(29,·)$, $\chi_{280}(69,·)$, $\chi_{280}(167,·)$, $\chi_{280}(41,·)$, $\chi_{280}(43,·)$, $\chi_{280}(181,·)$, $\chi_{280}(183,·)$, $\chi_{280}(111,·)$, $\chi_{280}(57,·)$, $\chi_{280}(267,·)$, $\chi_{280}(197,·)$, $\chi_{280}(71,·)$, $\chi_{280}(141,·)$, $\chi_{280}(209,·)$, $\chi_{280}(83,·)$, $\chi_{280}(223,·)$, $\chi_{280}(97,·)$, $\chi_{280}(99,·)$, $\chi_{280}(237,·)$, $\chi_{280}(239,·)$, $\chi_{280}(113,·)$, $\chi_{280}(211,·)$, $\chi_{280}(169,·)$, $\chi_{280}(251,·)$, $\chi_{280}(253,·)$, $\chi_{280}(127,·)$$\rbrace$ | ||
This is a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{12} + \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{13} + \frac{1}{6} a^{9} + \frac{1}{6} a^{5} + \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{5}{12} a^{14} + \frac{1}{12} a^{10} - \frac{5}{12} a^{6} + \frac{1}{12} a^{2}$, $\frac{1}{24} a^{19} + \frac{7}{24} a^{15} + \frac{1}{24} a^{11} + \frac{7}{24} a^{7} + \frac{1}{24} a^{3}$, $\frac{1}{10032} a^{20} - \frac{1}{48} a^{16} + \frac{17}{48} a^{12} - \frac{1}{48} a^{8} + \frac{17}{48} a^{4} + \frac{80}{209}$, $\frac{1}{20064} a^{21} - \frac{1}{96} a^{17} + \frac{17}{96} a^{13} - \frac{1}{96} a^{9} + \frac{17}{96} a^{5} + \frac{40}{209} a$, $\frac{1}{40128} a^{22} - \frac{1}{192} a^{18} - \frac{79}{192} a^{14} + \frac{95}{192} a^{10} + \frac{17}{192} a^{6} - \frac{169}{418} a^{2}$, $\frac{1}{80256} a^{23} - \frac{1}{384} a^{19} + \frac{113}{384} a^{15} - \frac{97}{384} a^{11} - \frac{175}{384} a^{7} - \frac{169}{836} a^{3}$, $\frac{1}{160512} a^{24} - \frac{1}{160512} a^{20} + \frac{11}{256} a^{16} - \frac{91}{256} a^{12} - \frac{85}{256} a^{8} - \frac{3313}{10032} a^{4} - \frac{224}{627}$, $\frac{1}{321024} a^{25} - \frac{1}{321024} a^{21} + \frac{11}{512} a^{17} - \frac{91}{512} a^{13} - \frac{85}{512} a^{9} - \frac{3313}{20064} a^{5} - \frac{112}{627} a$, $\frac{1}{642048} a^{26} - \frac{1}{642048} a^{22} + \frac{11}{1024} a^{18} - \frac{91}{1024} a^{14} - \frac{85}{1024} a^{10} + \frac{16751}{40128} a^{6} - \frac{56}{627} a^{2}$, $\frac{1}{1284096} a^{27} - \frac{1}{1284096} a^{23} + \frac{11}{2048} a^{19} + \frac{933}{2048} a^{15} + \frac{939}{2048} a^{11} + \frac{16751}{80256} a^{7} + \frac{571}{1254} a^{3}$, $\frac{1}{2568192} a^{28} - \frac{1}{2568192} a^{24} - \frac{5}{856064} a^{20} - \frac{529}{12288} a^{16} + \frac{4097}{12288} a^{12} + \frac{17845}{53504} a^{8} + \frac{3329}{10032} a^{4} + \frac{208}{627}$, $\frac{1}{5136384} a^{29} - \frac{1}{5136384} a^{25} - \frac{5}{1712128} a^{21} - \frac{529}{24576} a^{17} + \frac{4097}{24576} a^{13} + \frac{17845}{107008} a^{9} + \frac{3329}{20064} a^{5} + \frac{104}{627} a$, $\frac{1}{10272768} a^{30} - \frac{1}{10272768} a^{26} - \frac{5}{3424256} a^{22} - \frac{529}{49152} a^{18} - \frac{20479}{49152} a^{14} + \frac{17845}{214016} a^{10} - \frac{16735}{40128} a^{6} + \frac{52}{627} a^{2}$, $\frac{1}{20545536} a^{31} - \frac{1}{20545536} a^{27} - \frac{5}{6848512} a^{23} - \frac{529}{98304} a^{19} - \frac{20479}{98304} a^{15} - \frac{196171}{428032} a^{11} - \frac{16735}{80256} a^{7} - \frac{575}{1254} a^{3}$
Class group and class number
not computed
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( \frac{17}{321024} a^{29} + \frac{8641}{321024} a^{9} \) (order $40$) | 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: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_2^3\times C_4$ (as 32T34):
An abelian group of order 32 |
The 32 conjugacy class representatives for $C_2^3\times C_4$ |
Character table for $C_2^3\times C_4$ is not computed |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 | Data not computed | ||||||
$5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
$7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |