Normalized defining polynomial
\( x^{10} - 3x^{8} - 2x^{7} + 6x^{6} - 8x^{4} + 21x^{3} + 12x^{2} - 9x + 9 \)
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: | \(-25286677443\) \(\medspace = -\,3^{5}\cdot 101^{4}\) | 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: | \(10.97\) | 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: | $3^{1/2}101^{1/2}\approx 17.406895185529212$ | ||
Ramified primes: | \(3\), \(101\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{7}-\frac{1}{9}a^{6}-\frac{1}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{8}-\frac{1}{9}a^{6}-\frac{1}{9}a^{5}-\frac{2}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{225}a^{9}+\frac{1}{75}a^{8}+\frac{2}{75}a^{7}+\frac{16}{225}a^{6}-\frac{7}{75}a^{5}+\frac{4}{75}a^{4}-\frac{47}{225}a^{3}+\frac{7}{15}a^{2}-\frac{16}{75}a+\frac{8}{25}$
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: | \( \frac{2}{75} a^{9} + \frac{2}{25} a^{8} - \frac{13}{75} a^{7} - \frac{6}{25} a^{6} + \frac{8}{75} a^{5} + \frac{49}{75} a^{4} - \frac{44}{75} a^{3} + \frac{2}{15} a^{2} + \frac{43}{25} a - \frac{2}{25} \) (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: | $\frac{13}{225}a^{9}-\frac{11}{225}a^{8}-\frac{22}{225}a^{7}-\frac{17}{225}a^{6}+\frac{77}{225}a^{5}-\frac{44}{225}a^{4}-\frac{11}{225}a^{3}+\frac{7}{5}a^{2}-\frac{58}{75}a+\frac{4}{25}$, $\frac{11}{75}a^{9}+\frac{8}{75}a^{8}-\frac{127}{225}a^{7}-\frac{122}{225}a^{6}+\frac{23}{25}a^{5}+\frac{196}{225}a^{4}-\frac{451}{225}a^{3}+\frac{41}{15}a^{2}+\frac{322}{75}a-\frac{61}{25}$, $\frac{19}{225}a^{9}+\frac{7}{225}a^{8}-\frac{61}{225}a^{7}-\frac{71}{225}a^{6}+\frac{101}{225}a^{5}+\frac{103}{225}a^{4}-\frac{143}{225}a^{3}+\frac{23}{15}a^{2}+\frac{146}{75}a+\frac{2}{25}$, $\frac{1}{45}a^{9}-\frac{2}{45}a^{8}-\frac{4}{45}a^{7}+\frac{1}{45}a^{6}+\frac{14}{45}a^{5}-\frac{8}{45}a^{4}-\frac{17}{45}a^{3}+a^{2}-\frac{1}{15}a-\frac{7}{5}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 43.2958545685 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 43.2958545685 \cdot 1}{6\cdot\sqrt{25286677443}}\cr\approx \mathstrut & 0.444373879673 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.1.91809.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.6522683188340621511158049.1 |
Degree 10 sibling: | 10.2.851318140581.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{5}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 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 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(101\) | 101.2.0.1 | $x^{2} + 97 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
101.4.2.1 | $x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |