Normalized defining polynomial
\( x^{10} - 2x^{9} + 2x^{8} + x^{7} - 2x^{6} + 5x^{5} + 3x^{4} + 2x^{3} + 7x^{2} + 5x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2376404647\) \(\medspace = -\,47^{4}\cdot 487\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.66\) | 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: | $47^{1/2}487^{1/2}\approx 151.29111011556495$ | ||
Ramified primes: | \(47\), \(487\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-487}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $16a^{9}-39a^{8}+49a^{7}-5a^{6}-31a^{5}+95a^{4}+6a^{3}+29a^{2}+100a+36$, $10a^{9}-25a^{8}+33a^{7}-8a^{6}-14a^{5}+56a^{4}+2a^{3}+21a^{2}+59a+21$, $9a^{9}-23a^{8}+31a^{7}-9a^{6}-12a^{5}+51a^{4}-a^{3}+19a^{2}+52a+17$, $19a^{9}-47a^{8}+60a^{7}-9a^{6}-34a^{5}+111a^{4}+4a^{3}+36a^{2}+115a+39$ | 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: | \( 2.92841810966 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 2.92841810966 \cdot 1}{2\cdot\sqrt{2376404647}}\cr\approx \mathstrut & 0.294131971251 \end{aligned}\]
Galois group
$C_2\wr D_5$ (as 10T23):
A solvable group of order 320 |
The 20 conjugacy class representatives for $C_2\times (C_2^4 : D_5)$ |
Character table for $C_2\times (C_2^4 : D_5)$ |
Intermediate fields
5.1.2209.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(47\) | 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(487\) | $\Q_{487}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{487}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |