Normalized defining polynomial
\( x^{10} - x^{9} - 3x^{8} + 8x^{7} - 9x^{6} + x^{5} + 7x^{4} + 17x^{3} + 13x^{2} + 2x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(341108199621\) \(\medspace = 3^{5}\cdot 7^{5}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.23\) | 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: | $3^{1/2}7^{1/2}17^{1/2}\approx 18.894443627691185$ | ||
Ramified primes: | \(3\), \(7\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17}a^{8}-\frac{7}{17}a^{7}-\frac{4}{17}a^{6}-\frac{7}{17}a^{5}+\frac{1}{17}a^{4}+\frac{7}{17}a^{3}+\frac{7}{17}a^{2}-\frac{3}{17}a+\frac{2}{17}$, $\frac{1}{4301}a^{9}-\frac{114}{4301}a^{8}-\frac{530}{4301}a^{7}-\frac{1075}{4301}a^{6}-\frac{1239}{4301}a^{5}+\frac{147}{391}a^{4}+\frac{1723}{4301}a^{3}-\frac{378}{4301}a^{2}+\frac{28}{253}a-\frac{656}{4301}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{421}{4301}a^{9}-\frac{1189}{4301}a^{8}-\frac{237}{4301}a^{7}+\frac{315}{253}a^{6}-\frac{10559}{4301}a^{5}+\frac{845}{391}a^{4}-\frac{727}{4301}a^{3}+\frac{5059}{4301}a^{2}-\frac{8835}{4301}a-\frac{6225}{4301}$, $\frac{393}{4301}a^{9}-\frac{780}{4301}a^{8}-\frac{324}{4301}a^{7}+\frac{3577}{4301}a^{6}-\frac{7998}{4301}a^{5}+\frac{777}{391}a^{4}-\frac{3937}{4301}a^{3}+\frac{4764}{4301}a^{2}+\frac{3390}{4301}a+\frac{2276}{4301}$, $a$, $\frac{1855}{4301}a^{9}-\frac{1480}{4301}a^{8}-\frac{5811}{4301}a^{7}+\frac{13177}{4301}a^{6}-\frac{13502}{4301}a^{5}+\frac{89}{391}a^{4}+\frac{8112}{4301}a^{3}+\frac{2210}{253}a^{2}+\frac{33659}{4301}a+\frac{3086}{4301}$, $\frac{1583}{4301}a^{9}-\frac{1591}{4301}a^{8}-\frac{5102}{4301}a^{7}+\frac{12856}{4301}a^{6}-\frac{13490}{4301}a^{5}-\frac{105}{391}a^{4}+\frac{14084}{4301}a^{3}+\frac{25777}{4301}a^{2}+\frac{19049}{4301}a-\frac{1148}{4301}$ | 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: | \( 50.7666511079 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 50.7666511079 \cdot 1}{2\cdot\sqrt{341108199621}}\cr\approx \mathstrut & 0.270945476664 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{21}) \), 5.1.14161.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.33626538312268515533112249.2 |
Degree 10 sibling: | 10.0.828405627651.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{5}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.10.5.1 | $x^{10} + 162 x^{2} - 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |