Normalized defining polynomial
\( x^{10} - x^{8} - x^{7} - x^{5} + 2x^{4} + 3x^{3} - 3x^{2} + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(826110449\) \(\medspace = 1229\cdot 672181\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(7.79\) | 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: | $1229^{1/2}672181^{1/2}\approx 28742.137168276127$ | ||
Ramified primes: | \(1229\), \(672181\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{826110449}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{85}a^{9}+\frac{23}{85}a^{8}+\frac{18}{85}a^{7}-\frac{12}{85}a^{6}-\frac{21}{85}a^{5}+\frac{26}{85}a^{4}+\frac{1}{17}a^{3}+\frac{33}{85}a^{2}-\frac{9}{85}a-\frac{37}{85}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{64}{85}a^{9}+\frac{27}{85}a^{8}-\frac{38}{85}a^{7}-\frac{88}{85}a^{6}-\frac{69}{85}a^{5}-\frac{121}{85}a^{4}+\frac{13}{17}a^{3}+\frac{242}{85}a^{2}-\frac{66}{85}a+\frac{12}{85}$, $\frac{78}{85}a^{9}+\frac{9}{85}a^{8}-\frac{41}{85}a^{7}-\frac{86}{85}a^{6}-\frac{23}{85}a^{5}-\frac{97}{85}a^{4}+\frac{27}{17}a^{3}+\frac{194}{85}a^{2}-\frac{107}{85}a+\frac{4}{85}$, $\frac{63}{85}a^{9}+\frac{4}{85}a^{8}-\frac{56}{85}a^{7}-\frac{76}{85}a^{6}-\frac{48}{85}a^{5}-\frac{62}{85}a^{4}+\frac{29}{17}a^{3}+\frac{209}{85}a^{2}-\frac{142}{85}a+\frac{49}{85}$, $\frac{8}{17}a^{9}-\frac{3}{17}a^{8}-\frac{9}{17}a^{7}-\frac{11}{17}a^{6}+\frac{2}{17}a^{5}-\frac{13}{17}a^{4}+\frac{23}{17}a^{3}+\frac{26}{17}a^{2}-\frac{21}{17}a+\frac{10}{17}$, $\frac{2}{85}a^{9}+\frac{46}{85}a^{8}+\frac{36}{85}a^{7}-\frac{24}{85}a^{6}-\frac{42}{85}a^{5}-\frac{33}{85}a^{4}-\frac{15}{17}a^{3}-\frac{19}{85}a^{2}+\frac{152}{85}a-\frac{74}{85}$ | 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: | \( 1.6785661649701986 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 1.6785661649701986 \cdot 1}{2\cdot\sqrt{826110449}}\cr\approx \mathstrut & 0.182040859008242 \end{aligned}\]
Galois group
A non-solvable group of order 3628800 |
The 42 conjugacy class representatives for $S_{10}$ |
Character table for $S_{10}$ |
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 | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(1229\) | $\Q_{1229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(672181\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |