Normalized defining polynomial
\( x^{10} + 15x^{8} - 20x^{7} + 60x^{6} - 72x^{5} + 350x^{4} - 240x^{3} + 120x^{2} - 160x + 96 \)
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: | \(-17496000000000000\) \(\medspace = -\,2^{15}\cdot 3^{7}\cdot 5^{12}\) | 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: | \(42.10\) | 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: | $2^{11/4}3^{5/4}5^{271/200}\approx 235.1464422855153$ | ||
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: | \(\Q(\sqrt{-6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}-\frac{1}{5}$, $\frac{1}{30}a^{6}+\frac{1}{15}a^{5}-\frac{1}{6}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{7}{15}a-\frac{2}{5}$, $\frac{1}{60}a^{7}+\frac{1}{20}a^{5}+\frac{2}{5}a^{2}-\frac{1}{6}a+\frac{1}{5}$, $\frac{1}{360}a^{8}-\frac{1}{180}a^{7}+\frac{1}{120}a^{6}+\frac{1}{60}a^{5}-\frac{4}{15}a^{3}-\frac{89}{180}a^{2}-\frac{37}{90}a+\frac{1}{15}$, $\frac{1}{10800}a^{9}-\frac{1}{2700}a^{8}+\frac{31}{10800}a^{7}-\frac{1}{75}a^{6}+\frac{53}{900}a^{5}+\frac{71}{450}a^{4}+\frac{2167}{5400}a^{3}+\frac{503}{1350}a^{2}-\frac{647}{1350}a-\frac{22}{225}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{1}{900}a^{9}+\frac{17}{300}a^{8}+\frac{43}{450}a^{7}+\frac{277}{300}a^{6}+\frac{7}{300}a^{5}+\frac{359}{150}a^{4}-\frac{1823}{450}a^{3}+\frac{1199}{150}a^{2}-\frac{3613}{450}a+\frac{232}{75}$, $\frac{12347}{5400}a^{9}+\frac{5377}{5400}a^{8}+\frac{184577}{5400}a^{7}-\frac{6059}{200}a^{6}+\frac{52051}{450}a^{5}-\frac{21488}{225}a^{4}+\frac{1926929}{2700}a^{3}-\frac{461921}{2700}a^{2}-\frac{16984}{675}a-\frac{17623}{225}$, $\frac{7}{60}a^{9}+\frac{7}{40}a^{8}+\frac{97}{60}a^{7}-\frac{7}{8}a^{6}-\frac{9}{10}a^{5}-\frac{81}{5}a^{4}+\frac{91}{30}a^{3}-\frac{39}{20}a^{2}+\frac{22}{3}a-\frac{3}{5}$, $\frac{37}{2700}a^{9}-\frac{491}{5400}a^{8}+\frac{1027}{2700}a^{7}-\frac{1109}{600}a^{6}+\frac{1807}{450}a^{5}-\frac{4867}{450}a^{4}+\frac{8809}{1350}a^{3}-\frac{21317}{2700}a^{2}+\frac{1922}{675}a-\frac{886}{225}$ (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: | \( 95239.2044588 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 95239.2044588 \cdot 2}{2\cdot\sqrt{17496000000000000}}\cr\approx \mathstrut & 7.05091884147 \end{aligned}\] (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 | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
2.4.9.5 | $x^{4} + 10 x^{2} + 8 x + 6$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |