Normalized defining polynomial
\( x^{10} - 4x^{9} + 8x^{8} - 11x^{7} + 14x^{6} - 15x^{5} + 11x^{4} - 4x^{3} + x^{2} - x + 1 \)
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: | \(-562874607\) \(\medspace = -\,3^{2}\cdot 17^{2}\cdot 23\cdot 97^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.50\) | 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: | $3^{1/2}17^{1/2}23^{1/2}97^{1/2}\approx 337.31439340769316$ | ||
Ramified primes: | \(3\), \(17\), \(23\), \(97\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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: | $2a^{9}-6a^{8}+9a^{7}-10a^{6}+13a^{5}-11a^{4}+3a^{3}+2a^{2}-1$, $2a^{9}-7a^{8}+12a^{7}-15a^{6}+19a^{5}-19a^{4}+10a^{3}-2a^{2}+a-2$, $a^{9}-3a^{8}+5a^{7}-6a^{6}+8a^{5}-7a^{4}+4a^{3}+a$, $2a^{9}-6a^{8}+9a^{7}-10a^{6}+13a^{5}-11a^{4}+3a^{3}+2a^{2}+a-1$ | 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: | \( 1.07103275549 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1.07103275549 \cdot 1}{2\cdot\sqrt{562874607}}\cr\approx \mathstrut & 0.221037658245 \end{aligned}\]
Galois group
$C_2\wr S_5$ (as 10T39):
A non-solvable group of order 3840 |
The 36 conjugacy class representatives for $C_2 \wr S_5$ |
Character table for $C_2 \wr S_5$ |
Intermediate fields
5.1.1649.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 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 | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(17\) | 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.6.0.1 | $x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(23\) | $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
23.3.0.1 | $x^{3} + 2 x + 18$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(97\) | 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
97.4.2.1 | $x^{4} + 10474 x^{3} + 27919909 x^{2} + 2585716380 x + 183392493$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |