Normalized defining polynomial
\( x^{10} - 10x^{8} - 10x^{7} + 25x^{6} + 124x^{5} - 65x^{4} - 280x^{3} + 80x^{2} + 90x - 71 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-3690562500000000\) \(\medspace = -\,2^{8}\cdot 3^{10}\cdot 5^{12}\) | 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: | \(36.03\) | 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: | $2^{5/3}3^{37/18}5^{271/200}\approx 268.8872819707414$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{3}a^{3}-\frac{1}{6}a-\frac{1}{6}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a-\frac{1}{6}$, $\frac{1}{398202}a^{9}-\frac{10685}{398202}a^{8}+\frac{17975}{398202}a^{7}+\frac{153}{18962}a^{6}+\frac{7871}{66367}a^{5}+\frac{1071}{9481}a^{4}+\frac{43271}{132734}a^{3}-\frac{114619}{398202}a^{2}-\frac{165257}{398202}a+\frac{143467}{398202}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | 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: | $\frac{1199}{398202}a^{9}-\frac{414}{66367}a^{8}-\frac{2875}{66367}a^{7}+\frac{443}{56886}a^{6}+\frac{13215}{66367}a^{5}+\frac{4194}{9481}a^{4}-\frac{316663}{398202}a^{3}-\frac{85510}{66367}a^{2}+\frac{104337}{66367}a-\frac{6331}{398202}$, $\frac{2447}{398202}a^{9}+\frac{801}{132734}a^{8}-\frac{8248}{199101}a^{7}-\frac{2533}{28443}a^{6}-\frac{24646}{199101}a^{5}+\frac{3981}{9481}a^{4}+\frac{152683}{398202}a^{3}-\frac{138485}{398202}a^{2}-\frac{4874}{199101}a+\frac{24343}{199101}$, $\frac{1147}{18962}a^{9}+\frac{233}{56886}a^{8}-\frac{15256}{28443}a^{7}-\frac{13678}{28443}a^{6}+\frac{28606}{28443}a^{5}+\frac{56262}{9481}a^{4}-\frac{88923}{18962}a^{3}-\frac{213029}{18962}a^{2}+\frac{356578}{28443}a-\frac{140336}{28443}$, $\frac{5693}{398202}a^{9}-\frac{12511}{132734}a^{8}-\frac{12101}{66367}a^{7}+\frac{4129}{9481}a^{6}+\frac{78245}{66367}a^{5}+\frac{50165}{28443}a^{4}-\frac{1074373}{132734}a^{3}+\frac{219349}{132734}a^{2}+\frac{635732}{199101}a-\frac{873290}{199101}$, $\frac{2353}{199101}a^{9}+\frac{11288}{199101}a^{8}-\frac{27575}{398202}a^{7}-\frac{10019}{18962}a^{6}-\frac{108314}{199101}a^{5}+\frac{36107}{28443}a^{4}+\frac{275153}{66367}a^{3}-\frac{513955}{199101}a^{2}-\frac{246991}{132734}a+\frac{800615}{398202}$, $\frac{4108}{66367}a^{9}-\frac{9812}{199101}a^{8}-\frac{217393}{398202}a^{7}-\frac{16363}{56886}a^{6}+\frac{306469}{199101}a^{5}+\frac{188851}{28443}a^{4}-\frac{517977}{66367}a^{3}-\frac{578290}{66367}a^{2}+\frac{449143}{132734}a+\frac{605425}{398202}$ (assuming GRH) | 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: | \( 21174.4626058 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 21174.4626058 \cdot 1}{2\cdot\sqrt{3690562500000000}}\cr\approx \mathstrut & 0.691664851871 \end{aligned}\] (assuming GRH)
Galois group
$S_5\wr C_2$ (as 10T43):
A non-solvable group of order 28800 |
The 35 conjugacy class representatives for $S_5^2 \wr C_2$ |
Character table for $S_5^2 \wr C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 25 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Degree 36 sibling: | 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 | R | R | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ |
2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.6.10.4 | $x^{6} + 6 x^{3} + 81 x^{2} + 9$ | $3$ | $2$ | $10$ | $S_3^2$ | $[3/2, 5/2]_{2}^{2}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |