Normalized defining polynomial
\( x^{10} - x^{9} + 3x^{7} - 4x^{6} + 3x^{5} - 4x^{3} + 7x^{2} - 3x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-9464869683\) \(\medspace = -\,3^{5}\cdot 79^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(9.95\) | 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))
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Galois root discriminant: | $3^{1/2}79^{1/2}\approx 15.394804318340652$ | ||
Ramified primes: | \(3\), \(79\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17}a^{9}-\frac{3}{17}a^{8}+\frac{6}{17}a^{7}+\frac{8}{17}a^{6}-\frac{3}{17}a^{5}-\frac{8}{17}a^{4}-\frac{1}{17}a^{3}-\frac{2}{17}a^{2}-\frac{6}{17}a-\frac{8}{17}$
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)
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Torsion generator: | \( \frac{9}{17} a^{9} - \frac{10}{17} a^{8} + \frac{3}{17} a^{7} + \frac{21}{17} a^{6} - \frac{27}{17} a^{5} + \frac{30}{17} a^{4} - \frac{9}{17} a^{3} - \frac{18}{17} a^{2} + \frac{48}{17} a - \frac{4}{17} \) (order $6$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{1}{17}a^{9}-\frac{3}{17}a^{8}+\frac{6}{17}a^{7}-\frac{9}{17}a^{6}-\frac{3}{17}a^{5}+\frac{9}{17}a^{4}-\frac{18}{17}a^{3}+\frac{15}{17}a^{2}-\frac{6}{17}a+\frac{9}{17}$, $\frac{4}{17}a^{9}+\frac{5}{17}a^{8}-\frac{10}{17}a^{7}+\frac{15}{17}a^{6}+\frac{5}{17}a^{5}-\frac{15}{17}a^{4}+\frac{30}{17}a^{3}-\frac{25}{17}a^{2}+\frac{10}{17}a+\frac{19}{17}$, $\frac{3}{17}a^{9}-\frac{9}{17}a^{8}+\frac{1}{17}a^{7}+\frac{7}{17}a^{6}-\frac{26}{17}a^{5}+\frac{10}{17}a^{4}-\frac{3}{17}a^{3}-\frac{23}{17}a^{2}+\frac{33}{17}a-\frac{24}{17}$, $\frac{14}{17}a^{9}-\frac{8}{17}a^{8}-\frac{1}{17}a^{7}+\frac{44}{17}a^{6}-\frac{42}{17}a^{5}+\frac{24}{17}a^{4}+\frac{3}{17}a^{3}-\frac{62}{17}a^{2}+\frac{69}{17}a-\frac{27}{17}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 24.3676081141 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 24.3676081141 \cdot 1}{6\cdot\sqrt{9464869683}}\cr\approx \mathstrut & 0.408793210028 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.1.6241.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.559092234403032700371849.1 |
Degree 10 sibling: | 10.2.747724704957.1 |
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 | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(79\) | $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |