Normalized defining polynomial
\( x^{32} - 23 x^{28} + 528 x^{24} - 12121 x^{20} + 278255 x^{16} - 12121 x^{12} + 528 x^{8} - 23 x^{4} + 1 \)
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$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(517,·)$, $\chi_{840}(139,·)$, $\chi_{840}(13,·)$, $\chi_{840}(659,·)$, $\chi_{840}(533,·)$, $\chi_{840}(29,·)$, $\chi_{840}(799,·)$, $\chi_{840}(673,·)$, $\chi_{840}(167,·)$, $\chi_{840}(41,·)$, $\chi_{840}(811,·)$, $\chi_{840}(307,·)$, $\chi_{840}(181,·)$, $\chi_{840}(671,·)$, $\chi_{840}(701,·)$, $\chi_{840}(323,·)$, $\chi_{840}(197,·)$, $\chi_{840}(839,·)$, $\chi_{840}(713,·)$, $\chi_{840}(503,·)$, $\chi_{840}(463,·)$, $\chi_{840}(337,·)$, $\chi_{840}(169,·)$, $\chi_{840}(349,·)$, $\chi_{840}(827,·)$, $\chi_{840}(209,·)$, $\chi_{840}(491,·)$, $\chi_{840}(631,·)$, $\chi_{840}(377,·)$, $\chi_{840}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{12} + \frac{1}{5} a^{8} + \frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{13} + \frac{1}{5} a^{9} + \frac{1}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{14} + \frac{1}{5} a^{10} + \frac{1}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{15} + \frac{1}{5} a^{11} + \frac{1}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{1391275} a^{20} - \frac{290376}{1391275}$, $\frac{1}{1391275} a^{21} - \frac{290376}{1391275} a$, $\frac{1}{1391275} a^{22} - \frac{290376}{1391275} a^{2}$, $\frac{1}{1391275} a^{23} - \frac{290376}{1391275} a^{3}$, $\frac{1}{1391275} a^{24} - \frac{290376}{1391275} a^{4}$, $\frac{1}{1391275} a^{25} - \frac{290376}{1391275} a^{5}$, $\frac{1}{1391275} a^{26} - \frac{290376}{1391275} a^{6}$, $\frac{1}{1391275} a^{27} - \frac{290376}{1391275} a^{7}$, $\frac{1}{1391275} a^{28} - \frac{290376}{1391275} a^{8}$, $\frac{1}{1391275} a^{29} - \frac{290376}{1391275} a^{9}$, $\frac{1}{1391275} a^{30} - \frac{290376}{1391275} a^{10}$, $\frac{1}{1391275} a^{31} - \frac{290376}{1391275} a^{11}$
Class group and class number
$C_{4}\times C_{4}\times C_{4}\times C_{80}$, which has order $5120$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{1}{1391275} a^{22} - \frac{6665999}{1391275} a^{2} \) (order $20$) | 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: | \( 28949477018021.215 \) (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 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| $7$ | 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 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |