Normalized defining polynomial
\( x^{10} - 10x^{8} - 10x^{7} + 25x^{6} + 4x^{5} - 115x^{4} + 170x^{3} + 730x^{2} + 540x + 89 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[6, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2075941406250000\) \(\medspace = 2^{4}\cdot 3^{12}\cdot 5^{12}\) | 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: | \(34.02\) | 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: | $2^{2/3}3^{11/6}5^{271/200}\approx 105.32059888474208$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{5}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{5229416}a^{9}-\frac{579509}{5229416}a^{8}+\frac{200259}{5229416}a^{7}+\frac{614385}{5229416}a^{6}+\frac{67089}{653677}a^{5}+\frac{135257}{1307354}a^{4}-\frac{2266441}{5229416}a^{3}-\frac{1086983}{5229416}a^{2}+\frac{1898381}{5229416}a+\frac{1672487}{5229416}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | 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: | $\frac{25841}{2614708}a^{9}-\frac{1419}{653677}a^{8}-\frac{133995}{1307354}a^{7}-\frac{184191}{2614708}a^{6}+\frac{190890}{653677}a^{5}-\frac{34782}{653677}a^{4}-\frac{2872097}{2614708}a^{3}+\frac{1414020}{653677}a^{2}+\frac{8934279}{1307354}a+\frac{5457451}{2614708}$, $\frac{87851}{2614708}a^{9}-\frac{29842}{653677}a^{8}-\frac{381523}{1307354}a^{7}+\frac{226745}{2614708}a^{6}+\frac{567814}{653677}a^{5}-\frac{731476}{653677}a^{4}-\frac{6938215}{2614708}a^{3}+\frac{6502023}{653677}a^{2}+\frac{15735843}{1307354}a-\frac{7782685}{2614708}$, $\frac{25841}{1307354}a^{9}-\frac{2838}{653677}a^{8}-\frac{133995}{653677}a^{7}-\frac{184191}{1307354}a^{6}+\frac{381780}{653677}a^{5}-\frac{69564}{653677}a^{4}-\frac{2872097}{1307354}a^{3}+\frac{2828040}{653677}a^{2}+\frac{8280602}{653677}a+\frac{8072159}{1307354}$, $\frac{279955}{2614708}a^{9}-\frac{347465}{2614708}a^{8}-\frac{2367945}{2614708}a^{7}+\frac{21174}{653677}a^{6}+\frac{6895833}{2614708}a^{5}-\frac{7194377}{2614708}a^{4}-\frac{22983645}{2614708}a^{3}+\frac{38204127}{1307354}a^{2}+\frac{56356479}{1307354}a+\frac{4439097}{653677}$, $\frac{521587}{5229416}a^{9}-\frac{193921}{5229416}a^{8}-\frac{5093567}{5229416}a^{7}-\frac{3383485}{5229416}a^{6}+\frac{3494543}{1307354}a^{5}-\frac{1555763}{2614708}a^{4}-\frac{58900085}{5229416}a^{3}+\frac{107288125}{5229416}a^{2}+\frac{340453397}{5229416}a+\frac{181554327}{5229416}$, $\frac{39475}{5229416}a^{9}-\frac{37483}{5229416}a^{8}-\frac{345613}{5229416}a^{7}+\frac{123821}{5229416}a^{6}+\frac{517315}{2614708}a^{5}-\frac{308669}{653677}a^{4}-\frac{6831609}{5229416}a^{3}+\frac{13085249}{5229416}a^{2}+\frac{31127837}{5229416}a+\frac{10506157}{5229416}$, $\frac{559899}{5229416}a^{9}-\frac{857101}{5229416}a^{8}-\frac{4324031}{5229416}a^{7}+\frac{1255281}{5229416}a^{6}+\frac{1476237}{653677}a^{5}-\frac{2052250}{653677}a^{4}-\frac{36724687}{5229416}a^{3}+\frac{149085073}{5229416}a^{2}+\frac{177089291}{5229416}a+\frac{32803667}{5229416}$ (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: | \( 15921.2978384 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{2}\cdot 15921.2978384 \cdot 1}{2\cdot\sqrt{2075941406250000}}\cr\approx \mathstrut & 0.441449100373 \end{aligned}\] (assuming GRH)
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 | R | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
2.6.4.2 | $x^{6} - 2 x^{3} + 4$ | $3$ | $2$ | $4$ | $S_3\times C_3$ | $[\ ]_{3}^{6}$ | |
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.6.10.2 | $x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ | |
\(5\) | 5.10.12.15 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $[11/8, 11/8]_{8}^{2}$ |