Normalized defining polynomial
\( x^{10} - x^{9} + 2x^{8} - 4x^{7} + 5x^{6} - 7x^{5} + 9x^{4} - 8x^{3} + 5x^{2} - 4x + 1 \)
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: | \(3409076657\) \(\medspace = 7^{4}\cdot 17^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(8.98\) | 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: | $7^{1/2}17^{1/2}\approx 10.908712114635714$ | ||
Ramified primes: | \(7\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19}a^{9}+\frac{9}{19}a^{8}-\frac{3}{19}a^{7}+\frac{4}{19}a^{6}+\frac{7}{19}a^{5}+\frac{6}{19}a^{4}-\frac{7}{19}a^{3}-\frac{2}{19}a^{2}+\frac{4}{19}a-\frac{2}{19}$
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: | $\frac{11}{19}a^{9}-\frac{15}{19}a^{8}+\frac{24}{19}a^{7}-\frac{51}{19}a^{6}+\frac{58}{19}a^{5}-\frac{86}{19}a^{4}+\frac{113}{19}a^{3}-\frac{98}{19}a^{2}+\frac{63}{19}a-\frac{41}{19}$, $\frac{7}{19}a^{9}+\frac{6}{19}a^{8}-\frac{2}{19}a^{7}-\frac{10}{19}a^{6}-\frac{8}{19}a^{5}+\frac{4}{19}a^{4}-\frac{11}{19}a^{3}+\frac{24}{19}a^{2}-\frac{29}{19}a-\frac{14}{19}$, $\frac{14}{19}a^{9}-\frac{7}{19}a^{8}+\frac{15}{19}a^{7}-\frac{39}{19}a^{6}+\frac{41}{19}a^{5}-\frac{49}{19}a^{4}+\frac{73}{19}a^{3}-\frac{47}{19}a^{2}+\frac{18}{19}a-\frac{28}{19}$, $a^{9}-a^{8}+2a^{7}-4a^{6}+5a^{5}-7a^{4}+9a^{3}-8a^{2}+5a-4$, $\frac{9}{19}a^{9}-\frac{14}{19}a^{8}+\frac{30}{19}a^{7}-\frac{40}{19}a^{6}+\frac{63}{19}a^{5}-\frac{98}{19}a^{4}+\frac{108}{19}a^{3}-\frac{113}{19}a^{2}+\frac{74}{19}a-\frac{37}{19}$ | 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: | \( 4.02460808748 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4.02460808748 \cdot 1}{2\cdot\sqrt{3409076657}}\cr\approx \mathstrut & 0.214859559591 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 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: | 20.0.569468379011812486801.1 |
Degree 10 sibling: | 10.0.1403737447.1 |
Minimal sibling: | 10.0.1403737447.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{5}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
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 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |