Normalized defining polynomial
\( x^{10} - 5x^{9} + 5x^{8} - 10x^{7} - 5x^{6} + 243x^{5} - 705x^{4} + 2040x^{3} - 5040x^{2} + 5760x - 2304 \)
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: | \(-1640250000000000\) \(\medspace = -\,2^{10}\cdot 3^{8}\cdot 5^{12}\) | 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: | \(33.23\) | 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: | $2^{9/4}3^{25/18}5^{271/200}\approx 193.68244346412797$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{6}-\frac{1}{8}a^{5}+\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{24}a^{7}+\frac{1}{24}a^{6}+\frac{1}{12}a^{5}-\frac{7}{24}a^{4}-\frac{11}{24}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{96}a^{8}-\frac{1}{96}a^{7}-\frac{1}{32}a^{6}-\frac{1}{48}a^{5}-\frac{7}{32}a^{4}-\frac{29}{96}a^{3}-\frac{11}{32}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{24450432}a^{9}-\frac{25727}{8150144}a^{8}-\frac{215635}{24450432}a^{7}+\frac{645119}{12225216}a^{6}+\frac{3051499}{24450432}a^{5}+\frac{6111979}{24450432}a^{4}-\frac{4432937}{24450432}a^{3}-\frac{174153}{509384}a^{2}-\frac{13841}{63673}a-\frac{34107}{127346}$
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: | $\frac{171}{49696}a^{9}-\frac{557}{49696}a^{8}-\frac{565}{149088}a^{7}-\frac{182}{4659}a^{6}-\frac{11225}{149088}a^{5}+\frac{109057}{149088}a^{4}-\frac{172339}{149088}a^{3}+\frac{120435}{24848}a^{2}-\frac{27337}{3106}a+\frac{6179}{1553}$, $\frac{190099}{4075072}a^{9}-\frac{1749469}{12225216}a^{8}-\frac{508315}{12225216}a^{7}-\frac{3351337}{6112608}a^{6}-\frac{16091453}{12225216}a^{5}+\frac{107878931}{12225216}a^{4}-\frac{64709051}{4075072}a^{3}+\frac{32781201}{509384}a^{2}-\frac{27973567}{254692}a+\frac{3528641}{63673}$, $\frac{2153111}{24450432}a^{9}-\frac{9030379}{24450432}a^{8}+\frac{3162763}{24450432}a^{7}-\frac{9095335}{12225216}a^{6}-\frac{24559747}{24450432}a^{5}+\frac{169333695}{8150144}a^{4}-\frac{365499493}{8150144}a^{3}+\frac{35976609}{254692}a^{2}-\frac{20837867}{63673}a+\frac{28859101}{127346}$, $\frac{136395}{4075072}a^{9}-\frac{427443}{4075072}a^{8}-\frac{384455}{12225216}a^{7}-\frac{2123347}{6112608}a^{6}-\frac{11118133}{12225216}a^{5}+\frac{75505079}{12225216}a^{4}-\frac{145800677}{12225216}a^{3}+\frac{47009383}{1018768}a^{2}-\frac{9516293}{127346}a+\frac{2258592}{63673}$, $\frac{179003}{4075072}a^{9}-\frac{2135929}{12225216}a^{8}+\frac{583697}{12225216}a^{7}-\frac{2570515}{6112608}a^{6}-\frac{7429661}{12225216}a^{5}+\frac{121940399}{12225216}a^{4}-\frac{86645495}{4075072}a^{3}+\frac{71234377}{1018768}a^{2}-\frac{39168975}{254692}a+\frac{6649171}{63673}$, $\frac{744503}{24450432}a^{9}-\frac{735589}{8150144}a^{8}-\frac{131969}{24450432}a^{7}-\frac{4760705}{12225216}a^{6}-\frac{27439819}{24450432}a^{5}+\frac{126922817}{24450432}a^{4}-\frac{260852347}{24450432}a^{3}+\frac{90966519}{2037536}a^{2}-\frac{37016443}{509384}a+\frac{2274681}{63673}$ (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: | \( 15145.124628 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 15145.124628 \cdot 1}{2\cdot\sqrt{1640250000000000}}\cr\approx \mathstrut & 0.74207434994 \end{aligned}\] (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 | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.4.3 | $x^{4} + 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
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 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |