Normalized defining polynomial
\( x^{32} - x^{31} - 32 x^{30} + 31 x^{29} + 464 x^{28} - 433 x^{27} - 4033 x^{26} + 3600 x^{25} + 23426 x^{24} - 19826 x^{23} - 95978 x^{22} + 76152 x^{21} + 285340 x^{20} - 209188 x^{19} - 623732 x^{18} + 414544 x^{17} + 1004699 x^{16} - 590154 x^{15} - 1183593 x^{14} + 593422 x^{13} + 1001431 x^{12} - 407890 x^{11} - 589403 x^{10} + 181071 x^{9} + 228888 x^{8} - 46882 x^{7} - 53690 x^{6} + 5686 x^{5} + 6516 x^{4} - 116 x^{3} - 304 x^{2} - 16 x + 1 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[32, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(488368614066527220997452797221673658430576324462890625=5^{24}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $47.62$ | 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: | $5, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(85=5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{85}(1,·)$, $\chi_{85}(3,·)$, $\chi_{85}(4,·)$, $\chi_{85}(7,·)$, $\chi_{85}(9,·)$, $\chi_{85}(12,·)$, $\chi_{85}(16,·)$, $\chi_{85}(19,·)$, $\chi_{85}(21,·)$, $\chi_{85}(22,·)$, $\chi_{85}(23,·)$, $\chi_{85}(26,·)$, $\chi_{85}(27,·)$, $\chi_{85}(28,·)$, $\chi_{85}(36,·)$, $\chi_{85}(37,·)$, $\chi_{85}(48,·)$, $\chi_{85}(49,·)$, $\chi_{85}(57,·)$, $\chi_{85}(58,·)$, $\chi_{85}(59,·)$, $\chi_{85}(62,·)$, $\chi_{85}(63,·)$, $\chi_{85}(64,·)$, $\chi_{85}(66,·)$, $\chi_{85}(69,·)$, $\chi_{85}(73,·)$, $\chi_{85}(76,·)$, $\chi_{85}(78,·)$, $\chi_{85}(81,·)$, $\chi_{85}(82,·)$, $\chi_{85}(84,·)$$\rbrace$ | ||
| This is not 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
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $31$ | 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: | \( 34401098890034160 \) (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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ | $16^{2}$ | R | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{16}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 17 | Data not computed | ||||||