Normalized defining polynomial
\( x^{10} - 4x^{9} + x^{8} + 16x^{7} - 31x^{6} + 22x^{5} + 32x^{4} - 87x^{3} + 36x^{2} + 22x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(9932496465625\) \(\medspace = 5^{5}\cdot 19^{4}\cdot 29^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(19.94\) | 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: | $5^{1/2}19^{4/5}29^{3/4}\approx 294.6363877072366$ | ||
Ramified primes: | \(5\), \(19\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{145}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{939}a^{9}-\frac{61}{313}a^{8}-\frac{107}{939}a^{7}+\frac{389}{939}a^{6}-\frac{176}{939}a^{5}-\frac{400}{939}a^{4}+\frac{268}{939}a^{3}-\frac{170}{939}a^{2}+\frac{418}{939}a+\frac{320}{939}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | 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: | $\frac{167}{939}a^{9}-\frac{171}{313}a^{8}-\frac{28}{939}a^{7}+\frac{2050}{939}a^{6}-\frac{4039}{939}a^{5}+\frac{2686}{939}a^{4}+\frac{4379}{939}a^{3}-\frac{9610}{939}a^{2}+\frac{5015}{939}a+\frac{856}{939}$, $\frac{4}{313}a^{9}-\frac{106}{313}a^{8}+\frac{198}{313}a^{7}+\frac{304}{313}a^{6}-\frac{1017}{313}a^{5}+\frac{1217}{313}a^{4}-\frac{180}{313}a^{3}-\frac{3184}{313}a^{2}+\frac{2611}{313}a+\frac{967}{313}$, $\frac{103}{939}a^{9}-\frac{23}{313}a^{8}-\frac{692}{939}a^{7}+\frac{629}{939}a^{6}+\frac{652}{939}a^{5}-\frac{1762}{939}a^{4}+\frac{3190}{939}a^{3}+\frac{3148}{939}a^{2}-\frac{5774}{939}a-\frac{3661}{939}$, $\frac{343}{939}a^{9}-\frac{265}{313}a^{8}-\frac{1019}{939}a^{7}+\frac{3845}{939}a^{6}-\frac{4028}{939}a^{5}-\frac{106}{939}a^{4}+\frac{12109}{939}a^{3}-\frac{9482}{939}a^{2}-\frac{5927}{939}a+\frac{836}{939}$, $\frac{343}{939}a^{9}-\frac{265}{313}a^{8}-\frac{1019}{939}a^{7}+\frac{3845}{939}a^{6}-\frac{4028}{939}a^{5}-\frac{106}{939}a^{4}+\frac{12109}{939}a^{3}-\frac{9482}{939}a^{2}-\frac{5927}{939}a-\frac{103}{939}$, $\frac{343}{939}a^{9}-\frac{265}{313}a^{8}-\frac{1019}{939}a^{7}+\frac{3845}{939}a^{6}-\frac{4028}{939}a^{5}-\frac{106}{939}a^{4}+\frac{12109}{939}a^{3}-\frac{8543}{939}a^{2}-\frac{7805}{939}a+\frac{836}{939}$, $\frac{280}{939}a^{9}-\frac{178}{313}a^{8}-\frac{851}{939}a^{7}+\frac{2813}{939}a^{6}-\frac{3269}{939}a^{5}-\frac{259}{939}a^{4}+\frac{9310}{939}a^{3}-\frac{8162}{939}a^{2}-\frac{1274}{939}a+\frac{395}{939}$ | 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: | \( 970.442963618 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{2}\cdot 970.442963618 \cdot 1}{2\cdot\sqrt{9932496465625}}\cr\approx \mathstrut & 0.389000823656 \end{aligned}\]
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
\(\Q(\sqrt{5}) \) |
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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | ${\href{/padicField/23.10.0.1}{10} }$ | R | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
19.5.4.1 | $x^{5} + 19$ | $5$ | $1$ | $4$ | $D_{5}$ | $[\ ]_{5}^{2}$ | |
\(29\) | $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
29.4.3.4 | $x^{4} + 232$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |