Normalized defining polynomial
\( x^{10} - 2x^{9} + x^{8} - 16x^{7} + 64x^{6} - 104x^{5} + 84x^{4} - 32x^{3} + 12x^{2} - 8x + 4 \)
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: | \(-8388608000000\) \(\medspace = -\,2^{29}\cdot 5^{6}\) | 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: | \(19.61\) | 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: | $2^{63/16}5^{13/10}\approx 124.15568591030195$ | ||
Ramified primes: | \(2\), \(5\) | 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: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{8}+\frac{1}{4}a^{4}-\frac{1}{2}$, $\frac{1}{520}a^{9}+\frac{3}{26}a^{8}-\frac{49}{520}a^{7}-\frac{8}{65}a^{6}-\frac{1}{130}a^{5}+\frac{19}{260}a^{4}+\frac{23}{52}a^{3}-\frac{9}{65}a^{2}+\frac{57}{130}a-\frac{43}{130}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{29}{130} a^{9} - \frac{19}{52} a^{8} + \frac{9}{130} a^{7} - \frac{917}{260} a^{6} + \frac{852}{65} a^{5} - \frac{2343}{130} a^{4} + \frac{147}{13} a^{3} - \frac{463}{130} a^{2} + \frac{186}{65} a - \frac{89}{65} \) (order $4$) | 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: | $\frac{77}{130}a^{9}-\frac{25}{26}a^{8}+\frac{59}{260}a^{7}-\frac{1223}{130}a^{6}+\frac{8939}{260}a^{5}-\frac{3152}{65}a^{4}+\frac{825}{26}a^{3}-\frac{497}{65}a^{2}+\frac{461}{130}a-\frac{57}{65}$, $\frac{151}{520}a^{9}-\frac{43}{52}a^{8}+\frac{141}{520}a^{7}-\frac{298}{65}a^{6}+\frac{1452}{65}a^{5}-\frac{2484}{65}a^{4}+\frac{1445}{52}a^{3}-\frac{384}{65}a^{2}+\frac{157}{130}a-\frac{289}{65}$, $\frac{51}{520}a^{9}-\frac{3}{26}a^{8}-\frac{29}{520}a^{7}-\frac{397}{260}a^{6}+\frac{1263}{260}a^{5}-\frac{653}{130}a^{4}+\frac{55}{52}a^{3}+\frac{317}{130}a^{2}-\frac{139}{65}a+\frac{41}{65}$, $\frac{171}{65}a^{9}-\frac{255}{52}a^{8}+\frac{71}{65}a^{7}-\frac{2689}{65}a^{6}+\frac{21187}{130}a^{5}-\frac{61693}{260}a^{4}+\frac{1977}{13}a^{3}-\frac{1782}{65}a^{2}+\frac{963}{65}a-\frac{2989}{130}$ | 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: | \( 1839.9530215 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1839.9530215 \cdot 1}{4\cdot\sqrt{8388608000000}}\cr\approx \mathstrut & 1.5552533120 \end{aligned}\]
Galois group
$F_5\wr C_2$ (as 10T33):
A solvable group of order 800 |
The 20 conjugacy class representatives for $F_5 \wr C_2$ |
Character table for $F_5 \wr C_2$ |
Intermediate fields
\(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 25 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 | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.8.27.30 | $x^{8} + 8 x^{6} + 2 x^{4} + 2$ | $8$ | $1$ | $27$ | $C_4\wr C_2$ | $[2, 3, 7/2, 4, 9/2]$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.5.6.2 | $x^{5} + 15 x^{2} + 5$ | $5$ | $1$ | $6$ | $D_{5}$ | $[3/2]_{2}$ |