Normalized defining polynomial
\( x^{10} - 5x^{9} + 15x^{8} - 10x^{7} - 85x^{6} + 331x^{5} - 645x^{4} + 650x^{3} - 115x^{2} - 315x + 179 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(256000000000000\) \(\medspace = 2^{20}\cdot 5^{12}\) | 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: | \(27.59\) | 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: | $2^{3}5^{271/200}\approx 70.82626546363946$ | ||
Ramified primes: | \(2\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{47774}a^{9}-\frac{3053}{47774}a^{8}+\frac{6758}{23887}a^{7}-\frac{7795}{23887}a^{6}-\frac{16895}{47774}a^{5}-\frac{4081}{47774}a^{4}-\frac{3442}{23887}a^{3}+\frac{5148}{23887}a^{2}+\frac{5195}{47774}a-\frac{21481}{47774}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{21832}{23887}a^{9}-\frac{80027}{23887}a^{8}+\frac{220184}{23887}a^{7}+\frac{76644}{23887}a^{6}-\frac{1756224}{23887}a^{5}+\frac{4875066}{23887}a^{4}-\frac{7542776}{23887}a^{3}+\frac{4066392}{23887}a^{2}+\frac{2963752}{23887}a-\frac{2890048}{23887}$, $a-1$, $\frac{29133}{23887}a^{9}-\frac{107296}{23887}a^{8}+\frac{294964}{23887}a^{7}+\frac{99496}{23887}a^{6}-\frac{2351326}{23887}a^{5}+\frac{6538864}{23887}a^{4}-\frac{10124408}{23887}a^{3}+\frac{5474432}{23887}a^{2}+\frac{4010919}{23887}a-\frac{3884041}{23887}$, $\frac{10677}{23887}a^{9}-\frac{38900}{23887}a^{8}+\frac{104513}{23887}a^{7}+\frac{37960}{23887}a^{6}-\frac{853223}{23887}a^{5}+\frac{2314090}{23887}a^{4}-\frac{3607106}{23887}a^{3}+\frac{2176135}{23887}a^{2}+\frac{1315186}{23887}a-\frac{1542318}{23887}$, $\frac{5710595}{47774}a^{9}-\frac{10853677}{23887}a^{8}+\frac{29816481}{23887}a^{7}+\frac{7195298}{23887}a^{6}-\frac{468187019}{47774}a^{5}+\frac{664468317}{23887}a^{4}-\frac{1045004663}{23887}a^{3}+\frac{602890074}{23887}a^{2}+\frac{789833369}{47774}a-\frac{426151721}{23887}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 2534.34927274 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2534.34927274 \cdot 1}{2\cdot\sqrt{256000000000000}}\cr\approx \mathstrut & 0.493737318041 \end{aligned}\]
Galois group
$A_5^2:C_4$ (as 10T42):
A non-solvable group of order 14400 |
The 22 conjugacy class representatives for $A_5^2 : C_4$ |
Character table for $A_5^2 : C_4$ |
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 sibling: | 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 | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.20.18 | $x^{8} + 8 x^{7} + 28 x^{6} + 48 x^{5} + 68 x^{4} + 80 x^{3} + 168 x^{2} + 148$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 3, 3, 7/2]^{4}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |