Normalized defining polynomial
\( x^{10} - x^{9} + 3x^{8} + 2x^{6} + 3x^{5} + 6x^{4} - x^{3} + 10x^{2} - 3x + 1 \)
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: | \(-63215147763\) \(\medspace = -\,3^{5}\cdot 127^{4}\) | 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: | \(12.02\) | 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: | $3^{1/2}127^{1/2}\approx 19.519221295943137$ | ||
Ramified primes: | \(3\), \(127\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{295}a^{9}-\frac{102}{295}a^{8}-\frac{4}{59}a^{7}-\frac{9}{59}a^{6}+\frac{122}{295}a^{5}+\frac{71}{295}a^{4}-\frac{17}{59}a^{3}+\frac{29}{295}a^{2}+\frac{31}{295}a+\frac{111}{295}$
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: | \( -\frac{117}{295} a^{9} + \frac{134}{295} a^{8} - \frac{63}{59} a^{7} - \frac{9}{59} a^{6} - \frac{114}{295} a^{5} - \frac{342}{295} a^{4} - \frac{135}{59} a^{3} + \frac{147}{295} a^{2} - \frac{972}{295} a + \frac{288}{295} \) (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: | $\frac{17}{295}a^{9}+\frac{36}{295}a^{8}-\frac{9}{59}a^{7}+\frac{24}{59}a^{6}+\frac{9}{295}a^{5}+\frac{27}{295}a^{4}+\frac{6}{59}a^{3}+\frac{198}{295}a^{2}-\frac{358}{295}a+\frac{117}{295}$, $\frac{128}{295}a^{9}-\frac{76}{295}a^{8}+\frac{78}{59}a^{7}+\frac{28}{59}a^{6}+\frac{276}{295}a^{5}+\frac{533}{295}a^{4}+\frac{184}{59}a^{3}+\frac{172}{295}a^{2}+\frac{1313}{295}a+\frac{48}{295}$, $\frac{46}{295}a^{9}+\frac{28}{295}a^{8}-\frac{7}{59}a^{7}+\frac{58}{59}a^{6}+\frac{7}{295}a^{5}+\frac{21}{295}a^{4}+\frac{103}{59}a^{3}+\frac{449}{295}a^{2}-\frac{344}{295}a+\frac{386}{295}$, $\frac{6}{295}a^{9}-\frac{22}{295}a^{8}-\frac{24}{59}a^{7}+\frac{5}{59}a^{6}-\frac{153}{295}a^{5}-\frac{164}{295}a^{4}-\frac{43}{59}a^{3}-\frac{416}{295}a^{2}-\frac{404}{295}a-\frac{219}{295}$ | 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: | \( 71.0037805603 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 71.0037805603 \cdot 1}{6\cdot\sqrt{63215147763}}\cr\approx \mathstrut & 0.460913030029 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.1.16129.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.64453982490130814650341801.1 |
Degree 10 sibling: | 10.2.8028323765901.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{5}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(127\) | $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.2.1.2 | $x^{2} + 127$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |