Normalized defining polynomial
\( x^{10} - x^{9} + 3x^{8} + 6x^{7} + 9x^{6} + 3x^{5} + 3x^{4} + 66x^{3} + 75x^{2} - 91x + 169 \)
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: | \(-5910009391872\) \(\medspace = -\,2^{8}\cdot 3^{11}\cdot 19^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.93\) | 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: | $2^{4/5}3^{3/2}19^{1/2}\approx 39.43507572250456$ | ||
Ramified primes: | \(2\), \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{9}a^{5}+\frac{1}{9}a^{4}+\frac{1}{9}a^{3}+\frac{2}{9}a^{2}+\frac{2}{9}a+\frac{2}{9}$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{27}a^{8}+\frac{1}{27}a^{7}+\frac{1}{27}a^{6}+\frac{1}{27}a^{5}+\frac{1}{27}a^{4}+\frac{1}{27}a^{3}-\frac{2}{27}a^{2}-\frac{2}{27}a-\frac{2}{27}$, $\frac{1}{9477}a^{9}-\frac{22}{3159}a^{8}+\frac{1}{117}a^{7}+\frac{2}{3159}a^{6}-\frac{127}{3159}a^{5}-\frac{173}{1053}a^{4}-\frac{311}{3159}a^{3}+\frac{230}{3159}a^{2}+\frac{524}{1053}a-\frac{97}{729}$
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: | \( -\frac{80}{9477} a^{9} + \frac{5}{3159} a^{8} - \frac{2}{117} a^{7} - \frac{160}{3159} a^{6} - \frac{370}{3159} a^{5} + \frac{34}{1053} a^{4} + \frac{310}{3159} a^{3} - \frac{850}{3159} a^{2} - \frac{736}{1053} a + \frac{794}{729} \) (order $6$) | 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: | $\frac{5}{729}a^{9}-\frac{2}{243}a^{8}+\frac{10}{243}a^{6}-\frac{14}{243}a^{5}-\frac{10}{81}a^{4}-\frac{151}{243}a^{3}+\frac{16}{243}a^{2}-\frac{26}{81}a-\frac{797}{729}$, $\frac{68}{9477}a^{9}-\frac{92}{3159}a^{8}+\frac{1}{39}a^{7}+\frac{136}{3159}a^{6}-\frac{914}{3159}a^{5}+\frac{287}{1053}a^{4}-\frac{3247}{3159}a^{3}+\frac{6514}{3159}a^{2}-\frac{3212}{1053}a+\frac{208}{729}$, $\frac{16}{3159}a^{9}-\frac{1}{1053}a^{8}-\frac{4}{117}a^{7}+\frac{32}{1053}a^{6}+\frac{74}{1053}a^{5}-\frac{233}{351}a^{4}-\frac{62}{1053}a^{3}+\frac{170}{1053}a^{2}-\frac{313}{351}a-\frac{13}{243}$, $\frac{4}{1053}a^{9}+\frac{1}{117}a^{8}+\frac{4}{351}a^{7}-\frac{2}{39}a^{6}+\frac{4}{117}a^{5}+\frac{4}{351}a^{4}+\frac{95}{351}a^{3}-\frac{53}{117}a^{2}-\frac{17}{351}a+\frac{11}{81}$ | 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: | \( 1657.63006945 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1657.63006945 \cdot 1}{6\cdot\sqrt{5910009391872}}\cr\approx \mathstrut & 1.11286279173 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.3.1403568.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.3.1403568.1 |
Degree 6 sibling: | 6.0.113689008.4 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.3.1403568.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.9.9 | $x^{6} + 6 x^{4} + 21$ | $6$ | $1$ | $9$ | $C_6$ | $[2]_{2}$ | |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |