Normalized defining polynomial
\( x^{32} + 32 x^{30} + 463 x^{28} + 4004 x^{26} + 23051 x^{24} + 93128 x^{22} + 271214 x^{20} + 575832 x^{18} + 891311 x^{16} + 995824 x^{14} + 786798 x^{12} + 425128 x^{10} + 149106 x^{8} + 31248 x^{6} + 3396 x^{4} + 144 x^{2} + 1 \)
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: | \(151142765320815856472639544599622796222554813389533609984=2^{64}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $56.96$ | 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: | $2, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(136=2^{3}\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{136}(1,·)$, $\chi_{136}(133,·)$, $\chi_{136}(7,·)$, $\chi_{136}(9,·)$, $\chi_{136}(19,·)$, $\chi_{136}(23,·)$, $\chi_{136}(25,·)$, $\chi_{136}(29,·)$, $\chi_{136}(5,·)$, $\chi_{136}(33,·)$, $\chi_{136}(35,·)$, $\chi_{136}(37,·)$, $\chi_{136}(39,·)$, $\chi_{136}(43,·)$, $\chi_{136}(45,·)$, $\chi_{136}(49,·)$, $\chi_{136}(31,·)$, $\chi_{136}(61,·)$, $\chi_{136}(63,·)$, $\chi_{136}(67,·)$, $\chi_{136}(71,·)$, $\chi_{136}(79,·)$, $\chi_{136}(81,·)$, $\chi_{136}(83,·)$, $\chi_{136}(89,·)$, $\chi_{136}(95,·)$, $\chi_{136}(59,·)$, $\chi_{136}(109,·)$, $\chi_{136}(115,·)$, $\chi_{136}(121,·)$, $\chi_{136}(123,·)$, $\chi_{136}(125,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$
Class group and class number
$C_{2}\times C_{6}\times C_{2328}$, which has order $27936$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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: | \( 36938367173.77557 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{16}$ (as 32T32):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_{16}$ |
| Character table for $C_2\times C_{16}$ 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 | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
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 | Data not computed | ||||||
| 17 | Data not computed | ||||||