Normalized defining polynomial
\( x^{10} - 4x^{9} + 5x^{8} + 4x^{6} - 16x^{5} - 12x^{4} + 40x^{2} + 32x + 8 \)
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: | \(-41943040000000\) \(\medspace = -\,2^{29}\cdot 5^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(23.03\) | 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: | $2^{63/16}5^{31/20}\approx 185.65605360765468$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-10}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{1268}a^{9}-\frac{103}{1268}a^{8}-\frac{259}{1268}a^{7}+\frac{281}{1268}a^{6}-\frac{59}{317}a^{5}+\frac{131}{317}a^{4}+\frac{25}{317}a^{3}+\frac{61}{317}a^{2}-\frac{6}{317}a-\frac{32}{317}$
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: | \( \frac{37}{317} a^{9} - \frac{331}{634} a^{8} + \frac{244}{317} a^{7} - \frac{64}{317} a^{6} + \frac{144}{317} a^{5} - \frac{1483}{634} a^{4} - \frac{104}{317} a^{3} + \frac{152}{317} a^{2} + \frac{1648}{317} a + \frac{653}{317} \) (order $4$) | 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: | $\frac{11}{1268}a^{9}+\frac{135}{1268}a^{8}-\frac{313}{1268}a^{7}-\frac{79}{1268}a^{6}+\frac{302}{317}a^{5}+\frac{29}{634}a^{4}-\frac{42}{317}a^{3}-\frac{914}{317}a^{2}-\frac{1017}{317}a-\frac{35}{317}$, $\frac{153}{1268}a^{9}-\frac{543}{1268}a^{8}+\frac{315}{1268}a^{7}+\frac{1149}{1268}a^{6}-\frac{151}{317}a^{5}-\frac{562}{317}a^{4}-\frac{909}{634}a^{3}-\frac{177}{317}a^{2}+\frac{2569}{317}a+\frac{493}{317}$, $\frac{196}{317}a^{9}-\frac{3721}{1268}a^{8}+\frac{1541}{317}a^{7}-\frac{1913}{1268}a^{6}-\frac{291}{317}a^{5}-\frac{1272}{317}a^{4}-\frac{2907}{317}a^{3}+\frac{4078}{317}a^{2}+\frac{4489}{317}a+\frac{1223}{317}$, $\frac{1824}{317}a^{9}-\frac{15315}{634}a^{8}+\frac{21067}{634}a^{7}-\frac{2943}{634}a^{6}+\frac{6362}{317}a^{5}-\frac{29460}{317}a^{4}-\frac{33669}{634}a^{3}+\frac{6328}{317}a^{2}+\frac{70661}{317}a+\frac{43269}{317}$ | 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: | \( 2926.30902181 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 2926.30902181 \cdot 1}{4\cdot\sqrt{41943040000000}}\cr\approx \mathstrut & 1.10618963395 \end{aligned}\]
Galois group
$F_5\wr C_2$ (as 10T33):
A solvable group of order 800 |
The 20 conjugacy class representatives for $F_5 \wr C_2$ |
Character table for $F_5 \wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 25 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 | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.8.27.70 | $x^{8} + 8 x^{7} + 8 x^{6} + 2 x^{4} + 8 x^{2} + 18$ | $8$ | $1$ | $27$ | $C_4\wr C_2$ | $[2, 3, 7/2, 4, 9/2]$ | |
\(5\) | 5.5.7.4 | $x^{5} + 5 x^{3} + 5$ | $5$ | $1$ | $7$ | $F_5$ | $[7/4]_{4}$ |
5.5.0.1 | $x^{5} + 4 x + 3$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |