Normalized defining polynomial
\( x^{20} - 4 x^{19} + 66 x^{18} - 144 x^{17} + 2429 x^{16} - 4256 x^{15} + 66002 x^{14} + \cdots + 314911118323 \)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(73445842291921443780364922039304192\)
\(\medspace = 2^{55}\cdot 3^{10}\cdot 11^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(55.37\) | 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: | $2^{11/4}3^{1/2}11^{9/10}\approx 100.84317994603103$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{22}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19}a^{18}-\frac{4}{19}a^{16}-\frac{8}{19}a^{15}-\frac{2}{19}a^{14}+\frac{1}{19}a^{13}+\frac{7}{19}a^{12}+\frac{2}{19}a^{11}+\frac{2}{19}a^{10}+\frac{6}{19}a^{9}+\frac{7}{19}a^{8}-\frac{5}{19}a^{7}-\frac{8}{19}a^{6}+\frac{9}{19}a^{5}-\frac{6}{19}a^{4}+\frac{4}{19}a^{2}-\frac{7}{19}a+\frac{5}{19}$, $\frac{1}{10\!\cdots\!59}a^{19}+\frac{10\!\cdots\!90}{10\!\cdots\!59}a^{18}-\frac{17\!\cdots\!93}{10\!\cdots\!59}a^{17}+\frac{10\!\cdots\!44}{10\!\cdots\!59}a^{16}+\frac{29\!\cdots\!77}{10\!\cdots\!59}a^{15}+\frac{12\!\cdots\!54}{10\!\cdots\!59}a^{14}+\frac{25\!\cdots\!36}{10\!\cdots\!59}a^{13}-\frac{53\!\cdots\!43}{10\!\cdots\!59}a^{12}+\frac{44\!\cdots\!49}{10\!\cdots\!59}a^{11}+\frac{28\!\cdots\!49}{10\!\cdots\!59}a^{10}+\frac{38\!\cdots\!55}{10\!\cdots\!59}a^{9}+\frac{74\!\cdots\!04}{10\!\cdots\!59}a^{8}+\frac{49\!\cdots\!24}{10\!\cdots\!59}a^{7}+\frac{29\!\cdots\!90}{10\!\cdots\!59}a^{6}-\frac{15\!\cdots\!41}{10\!\cdots\!59}a^{5}+\frac{84\!\cdots\!66}{11\!\cdots\!47}a^{4}-\frac{69\!\cdots\!87}{10\!\cdots\!59}a^{3}+\frac{36\!\cdots\!83}{10\!\cdots\!59}a^{2}+\frac{11\!\cdots\!03}{10\!\cdots\!59}a-\frac{67\!\cdots\!39}{10\!\cdots\!59}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: | not computed | 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: | not computed | 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 $
Galois group
$C_{10}\wr C_2$ (as 20T53):
| A solvable group of order 200 |
| The 65 conjugacy class representatives for $C_{10}\wr C_2$ |
| Character table for $C_{10}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \), 4.0.202752.6, 10.0.479756288.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 20.0.5016449852600330836716407488512.2 |
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 | $20$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | $20$ | $20$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $20$ | $4$ | $5$ | $55$ | |||
|
\(3\)
| 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.10.5.1 | $x^{10} - 13310 x^{4} - 1449459$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 11.10.8.3 | $x^{10} - 110 x^{5} - 16819$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |