Normalized defining polynomial
\( x^{10} + 15x^{8} - 20x^{7} + 85x^{6} - 72x^{5} + 5x^{4} + 200x^{3} - 105x^{2} - 100x + 61 \)
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: | \(-5184000000000000\) \(\medspace = -\,2^{18}\cdot 3^{4}\cdot 5^{12}\) | 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: | \(37.28\) | 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: | $2^{9/4}3^{3/4}5^{271/200}\approx 95.99813307172106$ | ||
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{12}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}+\frac{5}{12}a+\frac{1}{3}$, $\frac{1}{1092}a^{8}+\frac{43}{1092}a^{7}+\frac{3}{26}a^{6}-\frac{45}{91}a^{5}+\frac{1}{39}a^{4}+\frac{19}{39}a^{3}-\frac{121}{1092}a^{2}+\frac{73}{364}a-\frac{107}{273}$, $\frac{1}{22932}a^{9}-\frac{5}{11466}a^{8}+\frac{425}{11466}a^{7}+\frac{17}{2548}a^{6}+\frac{2585}{11466}a^{5}+\frac{356}{819}a^{4}+\frac{389}{7644}a^{3}+\frac{293}{5733}a^{2}+\frac{1217}{11466}a+\frac{10399}{22932}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{17}{3276}a^{9}+\frac{97}{3276}a^{8}+\frac{269}{3276}a^{7}+\frac{57}{182}a^{6}-\frac{23}{126}a^{5}+\frac{353}{234}a^{4}-\frac{2641}{1092}a^{3}-\frac{599}{468}a^{2}+\frac{9215}{3276}a-\frac{1643}{1638}$, $\frac{11}{11466}a^{9}-\frac{73}{22932}a^{8}+\frac{89}{11466}a^{7}-\frac{29}{637}a^{6}-\frac{19}{11466}a^{5}+\frac{397}{1638}a^{4}-\frac{580}{1911}a^{3}-\frac{3469}{22932}a^{2}+\frac{3695}{11466}a-\frac{1153}{11466}$, $\frac{5654}{5733}a^{9}-\frac{11083}{11466}a^{8}+\frac{360371}{22932}a^{7}-\frac{89211}{2548}a^{6}+\frac{1348765}{11466}a^{5}-\frac{150926}{819}a^{4}+\frac{114053}{637}a^{3}+\frac{30413}{882}a^{2}-\frac{3492991}{22932}a+\frac{1359893}{22932}$, $\frac{691}{1274}a^{9}+\frac{135}{196}a^{8}+\frac{5492}{637}a^{7}-\frac{185}{2548}a^{6}+\frac{50721}{1274}a^{5}+\frac{1607}{91}a^{4}-\frac{1285}{98}a^{3}+\frac{308429}{2548}a^{2}+\frac{79097}{1274}a-\frac{53979}{2548}$ (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: | \( 23234.8572432 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 23234.8572432 \cdot 2}{2\cdot\sqrt{5184000000000000}}\cr\approx \mathstrut & 3.16014386204 \end{aligned}\] (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 | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.8.16.30 | $x^{8} - 4 x^{7} + 28 x^{6} - 16 x^{5} + 24 x^{4} + 88 x^{3} + 40 x^{2} + 16 x + 52$ | $4$ | $2$ | $16$ | $C_2^3: C_4$ | $[2, 2, 3]^{4}$ | |
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |