Normalized defining polynomial
\( x^{10} - x^{9} + 2x^{8} - x^{7} - 4x^{6} + x^{5} - 5x^{4} - 3x^{3} + x^{2} - x - 1 \)
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: | \(3303544037\) \(\medspace = 47^{4}\cdot 677\) | 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: | \(8.95\) | 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: | $47^{1/2}677^{1/2}\approx 178.37881040078724$ | ||
Ramified primes: | \(47\), \(677\) | 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: | \(\Q(\sqrt{677}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{47}a^{9}-\frac{4}{47}a^{8}+\frac{14}{47}a^{7}+\frac{4}{47}a^{6}-\frac{16}{47}a^{5}+\frac{2}{47}a^{4}-\frac{11}{47}a^{3}-\frac{17}{47}a^{2}+\frac{5}{47}a-\frac{16}{47}$
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: | $\frac{12}{47}a^{9}-\frac{1}{47}a^{8}+\frac{27}{47}a^{7}+\frac{1}{47}a^{6}-\frac{51}{47}a^{5}-\frac{23}{47}a^{4}-\frac{132}{47}a^{3}-\frac{63}{47}a^{2}-\frac{34}{47}a-\frac{51}{47}$, $\frac{21}{47}a^{9}-\frac{37}{47}a^{8}+\frac{59}{47}a^{7}-\frac{57}{47}a^{6}-\frac{54}{47}a^{5}+\frac{89}{47}a^{4}-\frac{137}{47}a^{3}+\frac{66}{47}a^{2}+\frac{11}{47}a-\frac{54}{47}$, $\frac{32}{47}a^{9}-\frac{34}{47}a^{8}+\frac{72}{47}a^{7}-\frac{60}{47}a^{6}-\frac{89}{47}a^{5}+\frac{17}{47}a^{4}-\frac{164}{47}a^{3}-\frac{27}{47}a^{2}-\frac{28}{47}a-\frac{42}{47}$, $\frac{7}{47}a^{9}-\frac{28}{47}a^{8}+\frac{51}{47}a^{7}-\frac{66}{47}a^{6}+\frac{29}{47}a^{5}+\frac{61}{47}a^{4}-\frac{77}{47}a^{3}+\frac{69}{47}a^{2}-\frac{12}{47}a-\frac{65}{47}$, $\frac{27}{47}a^{9}-\frac{14}{47}a^{8}+\frac{49}{47}a^{7}-\frac{33}{47}a^{6}-\frac{103}{47}a^{5}-\frac{40}{47}a^{4}-\frac{156}{47}a^{3}-\frac{36}{47}a^{2}+\frac{41}{47}a-\frac{56}{47}$ | 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: | \( 4.07637755834 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4.07637755834 \cdot 1}{2\cdot\sqrt{3303544037}}\cr\approx \mathstrut & 0.221072041049 \end{aligned}\]
Galois group
$C_2\wr D_5$ (as 10T23):
A solvable group of order 320 |
The 20 conjugacy class representatives for $C_2\times (C_2^4 : D_5)$ |
Character table for $C_2\times (C_2^4 : D_5)$ |
Intermediate fields
5.1.2209.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 10 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 40 siblings: | data not computed |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | R | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(47\) | 47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(677\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |