Normalized defining polynomial
\( x^{10} - 2x^{9} + 5x^{8} + 4x^{6} - 24x^{5} + 4x^{4} + 20x^{2} - 8x + 4 \)
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: |
\(-679477248000\)
\(\medspace = -\,2^{26}\cdot 3^{4}\cdot 5^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.25\) | 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: | $2^{3}3^{1/2}5^{3/4}\approx 46.331687411530766$ | ||
Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{7}+\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{120}a^{9}+\frac{1}{24}a^{7}+\frac{1}{12}a^{6}-\frac{1}{20}a^{5}+\frac{1}{5}a^{4}+\frac{11}{60}a^{3}+\frac{11}{30}a^{2}+\frac{2}{5}a-\frac{4}{15}$
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: |
\( -\frac{1}{6} a^{9} + \frac{1}{4} a^{8} - \frac{5}{6} a^{7} - \frac{1}{6} a^{6} - \frac{3}{2} a^{5} + \frac{13}{4} a^{4} + \frac{1}{3} a^{3} + \frac{5}{3} a^{2} - 3 a + \frac{5}{6} \)
(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: |
$\frac{13}{60}a^{9}-\frac{3}{8}a^{8}+\frac{13}{12}a^{7}+\frac{7}{24}a^{6}+\frac{19}{20}a^{5}-\frac{71}{20}a^{4}-\frac{7}{30}a^{3}-\frac{103}{60}a^{2}+\frac{19}{10}a+\frac{1}{15}$, $\frac{1}{30}a^{9}+\frac{1}{24}a^{7}+\frac{1}{3}a^{6}+\frac{7}{40}a^{5}-\frac{9}{20}a^{4}-\frac{91}{60}a^{3}+\frac{7}{15}a^{2}-\frac{3}{20}a+\frac{43}{30}$, $\frac{1}{24}a^{9}-\frac{1}{4}a^{8}+\frac{11}{24}a^{7}-\frac{5}{6}a^{6}-\frac{5}{2}a^{4}+\frac{41}{12}a^{3}+\frac{1}{3}a^{2}+\frac{5}{2}a-\frac{13}{3}$, $\frac{3}{40}a^{9}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}+\frac{5}{8}a^{6}-\frac{19}{20}a^{5}-\frac{29}{20}a^{4}-\frac{17}{20}a^{3}+\frac{71}{20}a^{2}-\frac{7}{5}a+\frac{3}{5}$
| 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: | \( 453.142676197 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 453.142676197 \cdot 1}{4\cdot\sqrt{679477248000}}\cr\approx \mathstrut & 1.34581962228 \end{aligned}\]
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 | R | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.8.24.72 | $x^{8} + 4 x^{6} + 2 x^{4} + 12 x^{2} + 8 x + 18$ | $8$ | $1$ | $24$ | $C_4\wr C_2$ | $[2, 3, 4]^{4}$ | |
\(3\)
| 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.8.4.2 | $x^{8} - 6 x^{6} + 162$ | $2$ | $4$ | $4$ | $C_8$ | $[\ ]_{2}^{4}$ | |
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.5.0.1 | $x^{5} + 4 x + 3$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |