Normalized defining polynomial
\( x^{10} - 3x^{9} - x^{8} + 10x^{7} - 2x^{6} - 3x^{5} + 8x^{4} - 3x^{3} - 6x^{2} + 4x - 1 \)
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: | \(2653290315584\) \(\medspace = 2^{6}\cdot 3461^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.47\) | 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: | $2^{2/3}3461^{1/2}\approx 93.38722346966051$ | ||
Ramified primes: | \(2\), \(3461\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{3461}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{295}a^{9}-\frac{116}{295}a^{8}+\frac{127}{295}a^{7}+\frac{114}{295}a^{6}+\frac{96}{295}a^{5}+\frac{64}{295}a^{4}-\frac{144}{295}a^{3}+\frac{44}{295}a^{2}+\frac{37}{295}a-\frac{47}{295}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $a^{9}-3a^{8}-a^{7}+10a^{6}-2a^{5}-3a^{4}+8a^{3}-3a^{2}-6a+4$, $\frac{136}{295}a^{9}-\frac{436}{295}a^{8}-\frac{133}{295}a^{7}+\frac{1639}{295}a^{6}-\frac{514}{295}a^{5}-\frac{1031}{295}a^{4}+\frac{1066}{295}a^{3}-\frac{211}{295}a^{2}-\frac{868}{295}a+\frac{393}{295}$, $\frac{52}{295}a^{9}-\frac{132}{295}a^{8}-\frac{181}{295}a^{7}+\frac{618}{295}a^{6}+\frac{272}{295}a^{5}-\frac{507}{295}a^{4}-\frac{113}{295}a^{3}-\frac{72}{295}a^{2}-\frac{436}{295}a-\frac{84}{295}$, $\frac{284}{295}a^{9}-\frac{789}{295}a^{8}-\frac{512}{295}a^{7}+\frac{2876}{295}a^{6}+\frac{124}{295}a^{5}-\frac{1294}{295}a^{4}+\frac{2174}{295}a^{3}-\frac{779}{295}a^{2}-\frac{1882}{295}a+\frac{517}{295}$, $\frac{91}{59}a^{9}-\frac{290}{59}a^{8}-\frac{66}{59}a^{7}+\frac{993}{59}a^{6}-\frac{291}{59}a^{5}-\frac{489}{59}a^{4}+\frac{702}{59}a^{3}-\frac{303}{59}a^{2}-\frac{645}{59}a+\frac{443}{59}$ | 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: | \( 284.834343252 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 284.834343252 \cdot 2}{2\cdot\sqrt{2653290315584}}\cr\approx \mathstrut & 1.09013286047 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_{6}$ |
Character table for $S_{6}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.2.165830644724.1, 6.2.55376.1 |
Degree 12 siblings: | deg 12, deg 12 |
Degree 15 siblings: | 15.3.146928604515779584.1, deg 15 |
Degree 20 siblings: | 20.4.7039949498771842313261056.1, deg 20, deg 20 |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.55376.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.9.6.1 | $x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$ | $3$ | $3$ | $6$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(3461\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |