Normalized defining polynomial
\( x^{32} - x^{31} + 2 x^{30} - 3 x^{29} + 5 x^{28} - 8 x^{27} + 13 x^{26} - 21 x^{25} + 34 x^{24} - 55 x^{23} + 89 x^{22} - 144 x^{21} + 233 x^{20} - 377 x^{19} + 610 x^{18} - 987 x^{17} + 1597 x^{16} + 987 x^{15} + 610 x^{14} + 377 x^{13} + 233 x^{12} + 144 x^{11} + 89 x^{10} + 55 x^{9} + 34 x^{8} + 21 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1 \)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(1250223652010309685753479160887484565582275390625\)\(\medspace = 5^{16}\cdot 17^{30}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $31.84$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $5, 17$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $32$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(85=5\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{85}(1,·)$, $\chi_{85}(4,·)$, $\chi_{85}(6,·)$, $\chi_{85}(9,·)$, $\chi_{85}(11,·)$, $\chi_{85}(14,·)$, $\chi_{85}(16,·)$, $\chi_{85}(19,·)$, $\chi_{85}(21,·)$, $\chi_{85}(24,·)$, $\chi_{85}(26,·)$, $\chi_{85}(29,·)$, $\chi_{85}(31,·)$, $\chi_{85}(36,·)$, $\chi_{85}(39,·)$, $\chi_{85}(41,·)$, $\chi_{85}(44,·)$, $\chi_{85}(46,·)$, $\chi_{85}(49,·)$, $\chi_{85}(54,·)$, $\chi_{85}(56,·)$, $\chi_{85}(59,·)$, $\chi_{85}(61,·)$, $\chi_{85}(64,·)$, $\chi_{85}(66,·)$, $\chi_{85}(69,·)$, $\chi_{85}(71,·)$, $\chi_{85}(74,·)$, $\chi_{85}(76,·)$, $\chi_{85}(79,·)$, $\chi_{85}(81,·)$, $\chi_{85}(84,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1597} a^{17} - \frac{610}{1597}$, $\frac{1}{1597} a^{18} - \frac{610}{1597} a$, $\frac{1}{1597} a^{19} - \frac{610}{1597} a^{2}$, $\frac{1}{1597} a^{20} - \frac{610}{1597} a^{3}$, $\frac{1}{1597} a^{21} - \frac{610}{1597} a^{4}$, $\frac{1}{1597} a^{22} - \frac{610}{1597} a^{5}$, $\frac{1}{1597} a^{23} - \frac{610}{1597} a^{6}$, $\frac{1}{1597} a^{24} - \frac{610}{1597} a^{7}$, $\frac{1}{1597} a^{25} - \frac{610}{1597} a^{8}$, $\frac{1}{1597} a^{26} - \frac{610}{1597} a^{9}$, $\frac{1}{1597} a^{27} - \frac{610}{1597} a^{10}$, $\frac{1}{1597} a^{28} - \frac{610}{1597} a^{11}$, $\frac{1}{1597} a^{29} - \frac{610}{1597} a^{12}$, $\frac{1}{1597} a^{30} - \frac{610}{1597} a^{13}$, $\frac{1}{1597} a^{31} - \frac{610}{1597} a^{14}$
Class group and class number
$C_{17}$, which has order $17$ (assuming GRH)
Unit group
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -\frac{13}{1597} a^{24} - \frac{46368}{1597} a^{7} \) (order $34$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| 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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 42597284566.379654 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
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 | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | 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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 17 | Data not computed | ||||||