Normalized defining polynomial
\( x^{10} + 6x^{8} - 4x^{7} + 12x^{6} - 14x^{5} + 10x^{4} - 12x^{3} + 4x^{2} + 1 \)
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)
| |
Discriminant: | \(-17606356992\) \(\medspace = -\,2^{12}\cdot 3^{5}\cdot 7^{2}\cdot 19^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.58\) | 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: | $2^{9/4}3^{1/2}7^{2/3}19^{2/3}\approx 214.67388287353924$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{8}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{8}a+\frac{1}{8}$
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: | \( -\frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{7}{8} a^{7} + \frac{3}{8} a^{6} - \frac{23}{8} a^{5} - \frac{3}{8} a^{4} - \frac{15}{8} a^{3} - \frac{5}{8} a^{2} + \frac{17}{8} a + \frac{7}{8} \) (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: | $\frac{1}{8}a^{9}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}-\frac{11}{8}a^{6}-\frac{17}{8}a^{5}-\frac{5}{8}a^{4}-\frac{17}{8}a^{3}+\frac{21}{8}a^{2}+\frac{7}{8}a+\frac{1}{8}$, $\frac{1}{8}a^{9}+\frac{7}{8}a^{8}+\frac{15}{8}a^{7}+\frac{37}{8}a^{6}+\frac{31}{8}a^{5}+\frac{35}{8}a^{4}-\frac{1}{8}a^{3}-\frac{19}{8}a^{2}-\frac{17}{8}a-\frac{7}{8}$, $\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+\frac{5}{2}a^{7}+\frac{1}{2}a^{6}+\frac{3}{2}a^{5}-\frac{3}{2}a^{4}-\frac{7}{2}a^{3}+\frac{1}{2}a^{2}-\frac{3}{2}a+\frac{5}{2}$, $\frac{9}{8}a^{9}+\frac{7}{8}a^{8}+\frac{55}{8}a^{7}+\frac{5}{8}a^{6}+\frac{95}{8}a^{5}-\frac{45}{8}a^{4}+\frac{47}{8}a^{3}-\frac{59}{8}a^{2}-\frac{1}{8}a-\frac{7}{8}$ | 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: | \( 32.1743323113 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 32.1743323113 \cdot 1}{6\cdot\sqrt{17606356992}}\cr\approx \mathstrut & 0.395751785339 \end{aligned}\]
Galois group
$\PGOPlus(4,5)$ (as 10T41):
A non-solvable group of order 14400 |
The 25 conjugacy class representatives for $(A_5^2 : C_2):C_2$ |
Character table for $(A_5^2 : C_2):C_2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \) |
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 sibling: | 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 | R | R | ${\href{/padicField/5.10.0.1}{10} }$ | R | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.12.4 | $x^{8} + 8 x^{6} - 16 x^{5} - 376 x^{4} - 128 x^{3} + 8928 x^{2} + 21056 x + 13008$ | $2$ | $4$ | $12$ | $C_2^3: C_4$ | $[2, 2, 3]^{4}$ | |
\(3\) | 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.3.2.3 | $x^{3} + 21$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.4.0.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(19\) | 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.3.2.3 | $x^{3} + 38$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |