Normalized defining polynomial
\( x^{10} + 6x^{8} - 2x^{7} + 14x^{6} - 2x^{5} + 11x^{4} + 7x^{3} + x^{2} + 10x - 1 \)
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: | \(148307253125\) \(\medspace = 5^{5}\cdot 83^{4}\) | 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: | \(13.10\) | 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: | $5^{1/2}83^{1/2}\approx 20.37154878746336$ | ||
Ramified primes: | \(5\), \(83\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{207}a^{9}+\frac{16}{207}a^{8}-\frac{14}{207}a^{7}+\frac{50}{207}a^{6}-\frac{83}{207}a^{5}-\frac{19}{207}a^{4}+\frac{52}{207}a^{3}+\frac{80}{207}a^{2}-\frac{10}{69}a+\frac{13}{207}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
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: | $\frac{32}{207}a^{9}-\frac{40}{207}a^{8}+\frac{173}{207}a^{7}-\frac{263}{207}a^{6}+\frac{449}{207}a^{5}-\frac{401}{207}a^{4}+\frac{215}{207}a^{3}+\frac{76}{207}a^{2}-\frac{44}{69}a+\frac{347}{207}$, $\frac{40}{207}a^{9}+\frac{19}{207}a^{8}+\frac{199}{207}a^{7}-\frac{1}{207}a^{6}+\frac{337}{207}a^{5}+\frac{137}{207}a^{4}+\frac{148}{207}a^{3}+\frac{371}{207}a^{2}-\frac{3}{23}a+\frac{175}{207}$, $\frac{20}{207}a^{9}+\frac{44}{207}a^{8}+\frac{134}{207}a^{7}+\frac{172}{207}a^{6}+\frac{203}{207}a^{5}+\frac{241}{207}a^{4}+\frac{5}{207}a^{3}-\frac{56}{207}a^{2}-\frac{62}{69}a-\frac{85}{207}$, $\frac{50}{207}a^{9}-\frac{28}{207}a^{8}+\frac{197}{207}a^{7}-\frac{260}{207}a^{6}+\frac{266}{207}a^{5}-\frac{191}{207}a^{4}-\frac{229}{207}a^{3}+\frac{205}{207}a^{2}-\frac{44}{23}a-\frac{40}{207}$, $\frac{14}{207}a^{9}+\frac{17}{207}a^{8}+\frac{11}{207}a^{7}+\frac{79}{207}a^{6}-\frac{127}{207}a^{5}+\frac{355}{207}a^{4}-\frac{307}{207}a^{3}+\frac{85}{207}a^{2}+\frac{67}{69}a-\frac{232}{207}$ | 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: | \( 46.1401263901 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 46.1401263901 \cdot 1}{2\cdot\sqrt{148307253125}}\cr\approx \mathstrut & 0.373462545783 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.172225.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.151523839718807162587890625.1 |
Degree 10 sibling: | 10.0.2461900401875.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(83\) | 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |