Normalized defining polynomial
\( x^{10} + 10x^{8} + 40x^{6} - 12x^{5} + 65x^{4} - 60x^{3} + 20x^{2} - 120x + 20 \)
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: | \(1640250000000000\) \(\medspace = 2^{10}\cdot 3^{8}\cdot 5^{12}\) | 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: | \(33.23\) | 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: | $2^{9/4}3^{4/5}5^{163/100}\approx 157.8830840055145$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{8}+\frac{1}{6}a^{7}-\frac{1}{6}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{12}a^{2}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{1992}a^{9}-\frac{1}{498}a^{8}-\frac{35}{498}a^{7}-\frac{32}{83}a^{6}+\frac{19}{83}a^{5}+\frac{13}{166}a^{4}-\frac{559}{1992}a^{3}+\frac{23}{249}a^{2}+\frac{223}{996}a-\frac{24}{83}$
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: | $\frac{77}{996}a^{9}+\frac{95}{498}a^{8}+\frac{337}{498}a^{7}+\frac{187}{166}a^{6}+\frac{291}{166}a^{5}+\frac{259}{166}a^{4}-\frac{713}{996}a^{3}-\frac{691}{249}a^{2}-\frac{1255}{498}a+\frac{39}{83}$, $\frac{26}{249}a^{9}-\frac{167}{996}a^{8}+\frac{95}{249}a^{7}-\frac{99}{83}a^{6}+\frac{51}{83}a^{5}-\frac{308}{83}a^{4}+\frac{406}{249}a^{3}-\frac{1531}{996}a^{2}+\frac{1387}{249}a-\frac{273}{166}$, $\frac{295}{996}a^{9}+\frac{157}{498}a^{8}+\frac{1013}{498}a^{7}+\frac{171}{166}a^{6}+\frac{757}{166}a^{5}-\frac{215}{166}a^{4}-\frac{1063}{996}a^{3}-\frac{2117}{249}a^{2}-\frac{449}{498}a+\frac{33}{83}$, $\frac{6}{83}a^{9}-\frac{61}{498}a^{8}+\frac{355}{498}a^{7}-\frac{175}{498}a^{6}+\frac{741}{166}a^{5}+\frac{129}{166}a^{4}+\frac{2339}{166}a^{3}+\frac{1403}{249}a^{2}+\frac{4459}{249}a-\frac{823}{249}$, $\frac{77}{498}a^{9}+\frac{431}{332}a^{8}-\frac{135}{166}a^{7}+\frac{5687}{498}a^{6}-\frac{995}{166}a^{5}+\frac{3921}{166}a^{4}-\frac{3469}{249}a^{3}+\frac{5157}{332}a^{2}-\frac{3766}{83}a+\frac{1879}{498}$ | 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: | \( 57946.6700139 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 57946.6700139 \cdot 1}{2\cdot\sqrt{1640250000000000}}\cr\approx \mathstrut & 4.45987749491 \end{aligned}\]
Galois group
A non-solvable group of order 1814400 |
The 24 conjugacy class representatives for $A_{10}$ |
Character table for $A_{10}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 45 sibling: | 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 | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.3.0.1}{3} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
2.4.4.4 | $x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$ | $2$ | $2$ | $4$ | $D_{4}$ | $[2, 2]^{2}$ | |
\(3\) | 3.5.4.1 | $x^{5} + 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ |
3.5.4.1 | $x^{5} + 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
\(5\) | 5.5.5.4 | $x^{5} + 10 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
5.5.7.3 | $x^{5} + 20 x^{3} + 5$ | $5$ | $1$ | $7$ | $F_5$ | $[7/4]_{4}$ |