Normalized defining polynomial
\( x^{10} - 5x^{7} + 10x^{6} + 9x^{5} - 10x^{4} - 5x^{3} - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(98876953125\) \(\medspace = 3^{4}\cdot 5^{13}\) | 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: | \(12.58\) | 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: | $3^{1/2}5^{13/10}\approx 14.035297835467638$ | ||
Ramified primes: | \(3\), \(5\) | 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: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{2}{9}a^{5}-\frac{4}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{9}a^{2}-\frac{2}{9}a+\frac{4}{9}$, $\frac{1}{117}a^{9}-\frac{5}{117}a^{8}+\frac{4}{39}a^{7}-\frac{1}{9}a^{6}-\frac{29}{117}a^{5}-\frac{5}{39}a^{4}+\frac{1}{3}a^{3}-\frac{31}{117}a^{2}+\frac{17}{39}a+\frac{44}{117}$
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)
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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)
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Fundamental units: | $\frac{8}{39}a^{9}-\frac{1}{39}a^{8}-\frac{8}{39}a^{7}-a^{6}+\frac{80}{39}a^{5}+\frac{101}{39}a^{4}-\frac{14}{3}a^{3}-\frac{22}{13}a^{2}+\frac{19}{13}a+\frac{9}{13}$, $a^{9}-5a^{6}+10a^{5}+9a^{4}-10a^{3}-5a^{2}$, $\frac{32}{117}a^{9}-\frac{4}{117}a^{8}-\frac{2}{39}a^{7}-\frac{14}{9}a^{6}+\frac{320}{117}a^{5}+\frac{29}{13}a^{4}-\frac{8}{3}a^{3}-\frac{290}{117}a^{2}-\frac{18}{13}a+\frac{4}{117}$, $\frac{68}{117}a^{9}-\frac{28}{117}a^{8}-\frac{1}{39}a^{7}-\frac{26}{9}a^{6}+\frac{797}{117}a^{5}+\frac{115}{39}a^{4}-\frac{25}{3}a^{3}+\frac{76}{117}a^{2}+\frac{4}{13}a-\frac{167}{117}$, $\frac{25}{117}a^{9}+\frac{31}{117}a^{8}-\frac{4}{39}a^{7}-\frac{10}{9}a^{6}+\frac{94}{117}a^{5}+\frac{200}{39}a^{4}-\frac{1}{3}a^{3}-\frac{580}{117}a^{2}-\frac{43}{39}a+\frac{47}{117}$ | 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: | \( 34.5229624611 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 34.5229624611 \cdot 1}{2\cdot\sqrt{98876953125}}\cr\approx \mathstrut & 0.342223625203 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.140625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.87989866733551025390625.1 |
Degree 10 sibling: | 10.0.59326171875.1 |
Minimal sibling: | 10.0.59326171875.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{5}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
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 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(5\) | 5.10.13.2 | $x^{10} + 10 x^{4} + 5$ | $10$ | $1$ | $13$ | $D_{10}$ | $[3/2]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.375.10t3.a.b | $2$ | $ 3 \cdot 5^{3}$ | 10.2.98876953125.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.375.5t2.a.b | $2$ | $ 3 \cdot 5^{3}$ | 5.1.140625.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.375.5t2.a.a | $2$ | $ 3 \cdot 5^{3}$ | 5.1.140625.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.375.10t3.a.a | $2$ | $ 3 \cdot 5^{3}$ | 10.2.98876953125.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |