Normalized defining polynomial
\( x^{10} - 3x^{9} + 8x^{8} - 10x^{7} + 10x^{6} - x^{5} - 5x^{4} + 7x^{3} - 2x^{2} - 3x + 3 \)
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: | \(-36409926927\) \(\medspace = -\,3\cdot 11^{5}\cdot 179\cdot 421\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.38\) | 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: | $3^{1/2}11^{1/2}179^{1/2}421^{1/2}\approx 1576.9740010539172$ | ||
Ramified primes: | \(3\), \(11\), \(179\), \(421\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-2486847}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$
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: | $\frac{1}{5}a^{9}+\frac{3}{5}a^{7}+\frac{4}{5}a^{6}+\frac{2}{5}a^{5}+2a^{4}+\frac{7}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{9}-a^{8}+\frac{8}{5}a^{7}-\frac{11}{5}a^{6}-\frac{3}{5}a^{5}+a^{4}-3a^{3}-\frac{3}{5}a^{2}+\frac{4}{5}a-\frac{1}{5}$, $\frac{2}{5}a^{9}-a^{8}+\frac{11}{5}a^{7}-\frac{7}{5}a^{6}+\frac{4}{5}a^{5}+2a^{4}-a^{3}+\frac{4}{5}a^{2}+\frac{3}{5}a-\frac{2}{5}$, $\frac{4}{5}a^{9}-2a^{8}+\frac{22}{5}a^{7}-\frac{19}{5}a^{6}+\frac{8}{5}a^{5}+2a^{4}-3a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{4}{5}$ | 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: | \( 21.7517591768 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 21.7517591768 \cdot 1}{2\cdot\sqrt{36409926927}}\cr\approx \mathstrut & 0.558153726709 \end{aligned}\]
Galois group
$S_5\wr C_2$ (as 10T43):
A non-solvable group of order 28800 |
The 35 conjugacy class representatives for $S_5^2 \wr C_2$ |
Character table for $S_5^2 \wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-11}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 25 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Degree 36 sibling: | 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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.5.0.1 | $x^{5} + 2 x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(11\) | 11.10.5.2 | $x^{10} + 55 x^{8} + 20 x^{7} + 1210 x^{6} - 202 x^{5} + 13410 x^{4} - 14080 x^{3} + 75585 x^{2} - 68970 x + 171252$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(179\) | $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.3.0.1 | $x^{3} + 4 x + 177$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(421\) | $\Q_{421}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |