Normalized defining polynomial
\( x^{10} - x^{9} + 6x^{8} - 8x^{7} + 8x^{6} - 3x^{5} + 8x^{4} - 8x^{3} + 6x^{2} - x + 1 \)
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: | \(-93637876851\) \(\medspace = -\,3^{9}\cdot 17\cdot 23^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.51\) | 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: | $3^{7/6}17^{1/2}23^{2/3}\approx 120.13912978968409$ | ||
Ramified primes: | \(3\), \(17\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-51}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{9}a^{8}+\frac{4}{9}a^{7}-\frac{2}{9}a^{6}-\frac{4}{9}a^{5}-\frac{1}{9}a^{4}-\frac{4}{9}a^{3}-\frac{2}{9}a^{2}+\frac{4}{9}a+\frac{1}{9}$, $\frac{1}{27}a^{9}-\frac{1}{3}a^{7}+\frac{4}{27}a^{6}-\frac{4}{9}a^{5}-\frac{1}{3}a^{4}-\frac{13}{27}a^{3}+\frac{4}{9}a^{2}-\frac{2}{9}a-\frac{13}{27}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{5}{9}a^{9}-\frac{2}{9}a^{8}+\frac{28}{9}a^{7}-\frac{7}{3}a^{6}+\frac{20}{9}a^{5}+\frac{11}{9}a^{4}+\frac{8}{3}a^{3}-\frac{17}{9}a^{2}+\frac{16}{9}a+\frac{5}{9}$, $a$, $\frac{17}{27}a^{9}-\frac{1}{9}a^{8}+\frac{35}{9}a^{7}-\frac{61}{27}a^{6}+\frac{35}{9}a^{5}-\frac{14}{9}a^{4}+\frac{88}{27}a^{3}-\frac{20}{9}a^{2}+\frac{25}{9}a-\frac{8}{27}$, $\frac{5}{27}a^{9}-\frac{1}{3}a^{8}+a^{7}-\frac{70}{27}a^{6}+\frac{10}{9}a^{5}-\frac{7}{3}a^{4}+\frac{25}{27}a^{3}-\frac{28}{9}a^{2}-\frac{4}{9}a-\frac{20}{27}$ | 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: | \( 23.0954688115 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 23.0954688115 \cdot 2}{2\cdot\sqrt{93637876851}}\cr\approx \mathstrut & 0.739095137688 \end{aligned}\]
Galois group
$C_2\wr A_5$ (as 10T36):
A non-solvable group of order 1920 |
The 24 conjugacy class representatives for $C_2 \wr A_5$ |
Character table for $C_2 \wr A_5$ |
Intermediate fields
5.1.42849.1 |
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 30 siblings: | data not computed |
Degree 32 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.7.1 | $x^{6} + 6 x^{2} + 6$ | $6$ | $1$ | $7$ | $S_3$ | $[3/2]_{2}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(23\) | 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
23.3.2.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
23.3.2.1 | $x^{3} + 23$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |