Normalized defining polynomial
\( x^{32} + 2207x^{16} + 1 \)
Invariants
Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(51922968585348276285304963292200960000000000000000\) \(\medspace = 2^{128}\cdot 5^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(35.78\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{4}5^{1/2}\approx 35.77708763999664$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(160=2^{5}\cdot 5\) | ||
Dirichlet character group: | $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(131,·)$, $\chi_{160}(129,·)$, $\chi_{160}(9,·)$, $\chi_{160}(11,·)$, $\chi_{160}(141,·)$, $\chi_{160}(19,·)$, $\chi_{160}(21,·)$, $\chi_{160}(151,·)$, $\chi_{160}(29,·)$, $\chi_{160}(159,·)$, $\chi_{160}(39,·)$, $\chi_{160}(41,·)$, $\chi_{160}(49,·)$, $\chi_{160}(51,·)$, $\chi_{160}(31,·)$, $\chi_{160}(61,·)$, $\chi_{160}(139,·)$, $\chi_{160}(69,·)$, $\chi_{160}(71,·)$, $\chi_{160}(79,·)$, $\chi_{160}(81,·)$, $\chi_{160}(89,·)$, $\chi_{160}(91,·)$, $\chi_{160}(99,·)$, $\chi_{160}(101,·)$, $\chi_{160}(109,·)$, $\chi_{160}(111,·)$, $\chi_{160}(59,·)$, $\chi_{160}(119,·)$, $\chi_{160}(121,·)$, $\chi_{160}(149,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{32768}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{987}a^{16}-\frac{377}{987}$, $\frac{1}{987}a^{17}-\frac{377}{987}a$, $\frac{1}{987}a^{18}-\frac{377}{987}a^{2}$, $\frac{1}{987}a^{19}-\frac{377}{987}a^{3}$, $\frac{1}{987}a^{20}-\frac{377}{987}a^{4}$, $\frac{1}{987}a^{21}-\frac{377}{987}a^{5}$, $\frac{1}{987}a^{22}-\frac{377}{987}a^{6}$, $\frac{1}{987}a^{23}-\frac{377}{987}a^{7}$, $\frac{1}{987}a^{24}-\frac{377}{987}a^{8}$, $\frac{1}{987}a^{25}-\frac{377}{987}a^{9}$, $\frac{1}{987}a^{26}-\frac{377}{987}a^{10}$, $\frac{1}{987}a^{27}-\frac{377}{987}a^{11}$, $\frac{1}{987}a^{28}-\frac{377}{987}a^{12}$, $\frac{1}{987}a^{29}-\frac{377}{987}a^{13}$, $\frac{1}{987}a^{30}-\frac{377}{987}a^{14}$, $\frac{1}{987}a^{31}-\frac{377}{987}a^{15}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{5}{987} a^{21} + \frac{10946}{987} a^{5} \) (order $32$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{1}{329}a^{21}+\frac{2255}{329}a^{5}$, $\frac{610}{987}a^{31}-\frac{89}{987}a^{27}+\frac{13}{987}a^{23}+\frac{1346269}{987}a^{15}-\frac{196418}{987}a^{11}+\frac{28657}{987}a^{7}$, $\frac{610}{987}a^{31}-\frac{34}{987}a^{25}+\frac{2}{987}a^{19}+\frac{1346269}{987}a^{15}-\frac{75025}{987}a^{9}+\frac{4181}{987}a^{3}$, $\frac{89}{987}a^{27}+\frac{34}{987}a^{25}+\frac{13}{987}a^{23}+\frac{196418}{987}a^{11}+\frac{75025}{987}a^{9}+\frac{28657}{987}a^{7}$, $\frac{48}{329}a^{28}-\frac{89}{987}a^{27}+\frac{55}{987}a^{26}+\frac{105937}{329}a^{12}-\frac{196418}{987}a^{11}+\frac{121393}{987}a^{10}$, $\frac{377}{987}a^{30}-\frac{34}{987}a^{25}+\frac{2}{987}a^{19}+\frac{832040}{987}a^{14}-\frac{75025}{987}a^{9}+\frac{4181}{987}a^{3}$, $\frac{34}{987}a^{25}+\frac{8}{987}a^{22}+\frac{2}{987}a^{19}+\frac{75025}{987}a^{9}+\frac{17711}{987}a^{6}+\frac{4181}{987}a^{3}$, $\frac{48}{329}a^{28}-\frac{55}{987}a^{26}+\frac{8}{987}a^{22}-\frac{2}{987}a^{19}+\frac{105937}{329}a^{12}-\frac{121393}{987}a^{10}+\frac{17711}{987}a^{6}-\frac{4181}{987}a^{3}+1$, $\frac{233}{987}a^{28}+\frac{1}{47}a^{24}+\frac{514229}{987}a^{12}+\frac{2208}{47}a^{8}-1$, $\frac{89}{987}a^{28}-\frac{8}{987}a^{22}+\frac{1}{987}a^{18}+\frac{196418}{987}a^{12}-\frac{17711}{987}a^{6}+\frac{2584}{987}a^{2}$, $\frac{610}{987}a^{30}+\frac{48}{329}a^{28}+\frac{1346269}{987}a^{14}+\frac{105937}{329}a^{12}-1$, $\frac{377}{987}a^{30}-\frac{377}{987}a^{29}+\frac{48}{329}a^{28}+\frac{832040}{987}a^{14}-\frac{832040}{987}a^{13}+\frac{105937}{329}a^{12}$, $\frac{48}{329}a^{28}-\frac{55}{987}a^{25}+\frac{8}{987}a^{22}+\frac{105937}{329}a^{12}-\frac{121393}{987}a^{9}+\frac{17711}{987}a^{6}$, $\frac{233}{987}a^{29}-\frac{13}{987}a^{23}-\frac{1}{987}a^{18}+\frac{514229}{987}a^{13}-\frac{28657}{987}a^{7}-\frac{1597}{987}a^{2}$, $\frac{89}{987}a^{28}-\frac{5}{987}a^{21}+\frac{2}{987}a^{19}+\frac{196418}{987}a^{12}-\frac{10946}{987}a^{5}+\frac{4181}{987}a^{3}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 323273529801.8212 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 323273529801.8212 \cdot 17}{32\cdot\sqrt{51922968585348276285304963292200960000000000000000}}\cr\approx \mathstrut & 0.140626518450650 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times C_8$ (as 32T37):
An abelian group of order 32 |
The 32 conjugacy class representatives for $C_2^2\times C_8$ |
Character table for $C_2^2\times C_8$ |
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 | ${\href{/padicField/3.8.0.1}{8} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.8.0.1}{8} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{16}$ | ${\href{/padicField/19.8.0.1}{8} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{16}$ | ${\href{/padicField/53.8.0.1}{8} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $32$ | $16$ | $2$ | $128$ | |||
\(5\) | 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |