Normalized defining polynomial
\( x^{10} - 2x^{9} + 4x^{8} - 5x^{7} + 7x^{6} - 4x^{5} + 7x^{4} - 3x^{3} + 5x^{2} - 2x + 1 \)
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)
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Discriminant: | \(-2304285743\) \(\medspace = -\,19^{2}\cdot 6383063\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.63\) | 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: | $19^{2/3}6383063^{1/2}\approx 17989.41190892089$ | ||
Ramified primes: | \(19\), \(6383063\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-6383063}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{29}a^{9}-\frac{10}{29}a^{8}-\frac{3}{29}a^{7}-\frac{10}{29}a^{6}-\frac{4}{29}a^{4}+\frac{10}{29}a^{3}+\frac{4}{29}a^{2}+\frac{2}{29}a+\frac{11}{29}$
Monogenic: | Not computed | |
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: | $a^{9}-2a^{8}+4a^{7}-5a^{6}+7a^{5}-4a^{4}+7a^{3}-3a^{2}+5a-2$, $\frac{28}{29}a^{9}-\frac{48}{29}a^{8}+\frac{90}{29}a^{7}-\frac{106}{29}a^{6}+5a^{5}-\frac{54}{29}a^{4}+\frac{164}{29}a^{3}-\frac{62}{29}a^{2}+\frac{85}{29}a-\frac{40}{29}$, $\frac{20}{29}a^{9}-\frac{55}{29}a^{8}+\frac{85}{29}a^{7}-\frac{113}{29}a^{6}+5a^{5}-\frac{109}{29}a^{4}+\frac{84}{29}a^{3}-\frac{123}{29}a^{2}+\frac{40}{29}a-\frac{41}{29}$, $\frac{14}{29}a^{9}-\frac{24}{29}a^{8}+\frac{45}{29}a^{7}-\frac{53}{29}a^{6}+3a^{5}-\frac{56}{29}a^{4}+\frac{111}{29}a^{3}-\frac{60}{29}a^{2}+\frac{86}{29}a-\frac{20}{29}$ | 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: | \( 3.5169195903949944 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 3.5169195903949944 \cdot 1}{2\cdot\sqrt{2304285743}}\cr\approx \mathstrut & 0.358726630802279 \end{aligned}\]
Galois group
A non-solvable group of order 3628800 |
The 42 conjugacy class representatives for $S_{10}$ |
Character table for $S_{10}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 20 sibling: | data not computed |
Degree 45 sibling: | 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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | 19.3.2.3 | $x^{3} + 38$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
19.7.0.1 | $x^{7} + 6 x + 17$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(6383063\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |