Normalized defining polynomial
\( x^{10} - 5x^{9} + 15x^{8} - 10x^{7} - 35x^{6} + 201x^{5} - 195x^{4} - 150x^{3} + 1035x^{2} + 135x - 711 \)
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: | \(-729000000000000\) \(\medspace = -\,2^{12}\cdot 3^{6}\cdot 5^{12}\) | 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: | \(30.64\) | 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))
| |
Galois root discriminant: | $2^{9/4}3^{25/18}5^{271/200}\approx 193.68244346412797$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{12}a^{7}+\frac{1}{12}a^{6}-\frac{1}{12}a^{4}+\frac{1}{3}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{24}a^{8}+\frac{1}{12}a^{6}+\frac{1}{12}a^{5}+\frac{5}{24}a^{4}-\frac{5}{12}a^{3}+\frac{1}{4}a+\frac{1}{8}$, $\frac{1}{360744}a^{9}-\frac{1345}{360744}a^{8}+\frac{2271}{60124}a^{7}-\frac{1402}{45093}a^{6}+\frac{9489}{120248}a^{5}+\frac{33161}{360744}a^{4}+\frac{80521}{180372}a^{3}-\frac{1899}{60124}a^{2}-\frac{50937}{120248}a-\frac{15177}{120248}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{1538}{45093}a^{9}-\frac{3121}{15031}a^{8}+\frac{11212}{15031}a^{7}-\frac{18248}{15031}a^{6}+\frac{11974}{45093}a^{5}+\frac{287024}{45093}a^{4}-\frac{629524}{45093}a^{3}+\frac{171076}{15031}a^{2}+\frac{301086}{15031}a-\frac{299703}{15031}$, $\frac{78}{15031}a^{9}-\frac{7663}{360744}a^{8}+\frac{1912}{45093}a^{7}+\frac{2831}{60124}a^{6}-\frac{64975}{180372}a^{5}+\frac{375137}{360744}a^{4}-\frac{33591}{60124}a^{3}-\frac{48899}{30062}a^{2}+\frac{226765}{60124}a+\frac{373021}{120248}$, $\frac{66137}{360744}a^{9}-\frac{33887}{30062}a^{8}+\frac{364291}{90186}a^{7}-\frac{581879}{90186}a^{6}+\frac{329725}{360744}a^{5}+\frac{1620950}{45093}a^{4}-\frac{1161628}{15031}a^{3}+\frac{921867}{15031}a^{2}+\frac{14357111}{120248}a-\frac{3398531}{30062}$, $\frac{2668}{45093}a^{9}-\frac{59359}{180372}a^{8}+\frac{202181}{180372}a^{7}-\frac{43471}{30062}a^{6}-\frac{27301}{60124}a^{5}+\frac{170712}{15031}a^{4}-\frac{3296105}{180372}a^{3}+\frac{607591}{60124}a^{2}+\frac{1358531}{30062}a-\frac{283033}{30062}$, $\frac{68701}{180372}a^{9}-\frac{107039}{45093}a^{8}+\frac{776185}{90186}a^{7}-\frac{2571533}{180372}a^{6}+\frac{337145}{90186}a^{5}+\frac{6533627}{90186}a^{4}-\frac{29447209}{180372}a^{3}+\frac{4184499}{30062}a^{2}+\frac{3446956}{15031}a-\frac{14193933}{60124}$, $\frac{12359}{30062}a^{9}-\frac{457375}{180372}a^{8}+\frac{548967}{60124}a^{7}-\frac{665357}{45093}a^{6}+\frac{514363}{180372}a^{5}+\frac{7159415}{90186}a^{4}-\frac{31169861}{180372}a^{3}+\frac{8446377}{60124}a^{2}+\frac{3957607}{15031}a-\frac{7562675}{30062}$ | 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: | \( 10185.7958438 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 10185.7958438 \cdot 1}{2\cdot\sqrt{729000000000000}}\cr\approx \mathstrut & 0.748618913522 \end{aligned}\]
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 | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.6.6 | $x^{4} - 4 x^{3} + 28 x^{2} - 24 x + 36$ | $2$ | $2$ | $6$ | $D_{4}$ | $[2, 3]^{2}$ | |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
\(3\) | 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |