Normalized defining polynomial
\( x^{32} - 76 x^{28} + 5170 x^{24} - 45134 x^{20} + 332199 x^{16} - 279214 x^{12} + 211915 x^{8} - 461 x^{4} + 1 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.29$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(43,·)$, $\chi_{840}(139,·)$, $\chi_{840}(533,·)$, $\chi_{840}(281,·)$, $\chi_{840}(799,·)$, $\chi_{840}(419,·)$, $\chi_{840}(167,·)$, $\chi_{840}(169,·)$, $\chi_{840}(811,·)$, $\chi_{840}(433,·)$, $\chi_{840}(181,·)$, $\chi_{840}(827,·)$, $\chi_{840}(629,·)$, $\chi_{840}(449,·)$, $\chi_{840}(323,·)$, $\chi_{840}(197,·)$, $\chi_{840}(71,·)$, $\chi_{840}(713,·)$, $\chi_{840}(503,·)$, $\chi_{840}(461,·)$, $\chi_{840}(547,·)$, $\chi_{840}(727,·)$, $\chi_{840}(349,·)$, $\chi_{840}(223,·)$, $\chi_{840}(97,·)$, $\chi_{840}(239,·)$, $\chi_{840}(757,·)$, $\chi_{840}(631,·)$, $\chi_{840}(377,·)$, $\chi_{840}(251,·)$, $\chi_{840}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19} a^{20} + \frac{8}{19} a^{16} + \frac{7}{19} a^{12} - \frac{8}{19} a^{8} - \frac{7}{19} a^{4} + \frac{1}{19}$, $\frac{1}{19} a^{21} + \frac{8}{19} a^{17} + \frac{7}{19} a^{13} - \frac{8}{19} a^{9} - \frac{7}{19} a^{5} + \frac{1}{19} a$, $\frac{1}{19} a^{22} + \frac{8}{19} a^{18} + \frac{7}{19} a^{14} - \frac{8}{19} a^{10} - \frac{7}{19} a^{6} + \frac{1}{19} a^{2}$, $\frac{1}{19} a^{23} + \frac{8}{19} a^{19} + \frac{7}{19} a^{15} - \frac{8}{19} a^{11} - \frac{7}{19} a^{7} + \frac{1}{19} a^{3}$, $\frac{1}{165509} a^{24} + \frac{648}{165509} a^{20} + \frac{37256}{165509} a^{16} + \frac{48039}{165509} a^{12} + \frac{17844}{165509} a^{8} + \frac{75416}{165509} a^{4} - \frac{78761}{165509}$, $\frac{1}{165509} a^{25} + \frac{648}{165509} a^{21} + \frac{37256}{165509} a^{17} + \frac{48039}{165509} a^{13} + \frac{17844}{165509} a^{9} + \frac{75416}{165509} a^{5} - \frac{78761}{165509} a$, $\frac{1}{165509} a^{26} + \frac{648}{165509} a^{22} + \frac{37256}{165509} a^{18} + \frac{48039}{165509} a^{14} + \frac{17844}{165509} a^{10} + \frac{75416}{165509} a^{6} - \frac{78761}{165509} a^{2}$, $\frac{1}{165509} a^{27} + \frac{648}{165509} a^{23} + \frac{37256}{165509} a^{19} + \frac{48039}{165509} a^{15} + \frac{17844}{165509} a^{11} + \frac{75416}{165509} a^{7} - \frac{78761}{165509} a^{3}$, $\frac{1}{3130324642056393739} a^{28} - \frac{2234010878325}{3130324642056393739} a^{24} + \frac{65872439905654818}{3130324642056393739} a^{20} - \frac{181479732487623801}{3130324642056393739} a^{16} + \frac{1229637479269120940}{3130324642056393739} a^{12} - \frac{229809350472950975}{3130324642056393739} a^{8} - \frac{76047805942701046}{3130324642056393739} a^{4} + \frac{1513097126457586883}{3130324642056393739}$, $\frac{1}{3130324642056393739} a^{29} - \frac{2234010878325}{3130324642056393739} a^{25} + \frac{65872439905654818}{3130324642056393739} a^{21} - \frac{181479732487623801}{3130324642056393739} a^{17} + \frac{1229637479269120940}{3130324642056393739} a^{13} - \frac{229809350472950975}{3130324642056393739} a^{9} - \frac{76047805942701046}{3130324642056393739} a^{5} + \frac{1513097126457586883}{3130324642056393739} a$, $\frac{1}{3130324642056393739} a^{30} - \frac{2234010878325}{3130324642056393739} a^{26} + \frac{65872439905654818}{3130324642056393739} a^{22} - \frac{181479732487623801}{3130324642056393739} a^{18} + \frac{1229637479269120940}{3130324642056393739} a^{14} - \frac{229809350472950975}{3130324642056393739} a^{10} - \frac{76047805942701046}{3130324642056393739} a^{6} + \frac{1513097126457586883}{3130324642056393739} a^{2}$, $\frac{1}{3130324642056393739} a^{31} - \frac{2234010878325}{3130324642056393739} a^{27} + \frac{65872439905654818}{3130324642056393739} a^{23} - \frac{181479732487623801}{3130324642056393739} a^{19} + \frac{1229637479269120940}{3130324642056393739} a^{15} - \frac{229809350472950975}{3130324642056393739} a^{11} - \frac{76047805942701046}{3130324642056393739} a^{7} + \frac{1513097126457586883}{3130324642056393739} a^{3}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{16}$, which has order $512$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{146004694177063315}{3130324642056393739} a^{30} + \frac{357946442848886980}{100978214259883669} a^{26} - \frac{754842987672817471835}{3130324642056393739} a^{22} + \frac{6589688810841769578110}{3130324642056393739} a^{18} - \frac{48501911780352420893085}{3130324642056393739} a^{14} + \frac{40761470733398916039397}{3130324642056393739} a^{10} - \frac{30940132199426449530685}{3130324642056393739} a^{6} + \frac{67307179500830824115}{3130324642056393739} a^{2} \) (order $12$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 28764339921106.887 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
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 | R | 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 | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 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 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |