Normalized defining polynomial
\( x^{10} - 5x^{9} + 25x^{8} - 60x^{7} + 120x^{6} - 116x^{5} + 20x^{4} + 80x^{3} - 160x^{2} + 80x - 16 \)
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))
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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)))
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{10768}a^{9}+\frac{109}{10768}a^{8}+\frac{337}{10768}a^{7}-\frac{169}{2692}a^{6}-\frac{111}{5384}a^{5}-\frac{299}{2692}a^{4}-\frac{431}{2692}a^{3}-\frac{329}{1346}a^{2}+\frac{81}{673}a-\frac{183}{673}$
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)
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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: | $\frac{21}{1346}a^{9}-\frac{133}{2692}a^{8}+\frac{347}{1346}a^{7}-\frac{799}{2692}a^{6}+\frac{361}{673}a^{5}+\frac{1589}{2692}a^{4}-\frac{604}{673}a^{3}-\frac{43}{673}a^{2}+\frac{148}{673}a-\frac{459}{673}$, $\frac{399}{10768}a^{9}-\frac{2273}{10768}a^{8}+\frac{10631}{10768}a^{7}-\frac{6861}{2692}a^{6}+\frac{24357}{5384}a^{5}-\frac{11621}{2692}a^{4}-\frac{2373}{2692}a^{3}+\frac{2674}{673}a^{2}-\frac{658}{673}a+\frac{340}{673}$, $\frac{271}{2692}a^{9}-\frac{2165}{5384}a^{8}+\frac{11039}{5384}a^{7}-\frac{19797}{5384}a^{6}+\frac{18579}{2692}a^{5}-\frac{942}{673}a^{4}-\frac{8147}{1346}a^{3}+\frac{5411}{673}a^{2}-\frac{7089}{673}a-\frac{1856}{673}$, $\frac{49}{1346}a^{9}-\frac{43}{1346}a^{8}+\frac{361}{1346}a^{7}+\frac{1199}{1346}a^{6}-\frac{2129}{1346}a^{5}+\frac{3675}{673}a^{4}-\frac{3043}{1346}a^{3}-\frac{11193}{1346}a^{2}+\frac{794}{673}a+\frac{275}{673}$, $\frac{417}{10768}a^{9}-\frac{3003}{10768}a^{8}+\frac{15351}{10768}a^{7}-\frac{25863}{5384}a^{6}+\frac{7590}{673}a^{5}-\frac{54691}{2692}a^{4}+\frac{14797}{673}a^{3}-\frac{25475}{1346}a^{2}+\frac{10349}{1346}a-\frac{935}{673}$ | 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: | \( 2615.79908463 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2615.79908463 \cdot 1}{2\cdot\sqrt{256000000000000}}\cr\approx \mathstrut & 0.509605222323 \end{aligned}\]
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 | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.8.20.19 | $x^{8} + 8 x^{7} + 12 x^{6} - 16 x^{5} + 20 x^{4} + 16 x^{3} + 40 x^{2} + 52$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 3, 3, 7/2]^{4}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |