Normalized defining polynomial
\( x^{10} - 4x^{9} + 6x^{8} - x^{7} - 26x^{6} + 29x^{5} - 3x^{4} - 76x^{3} - 21x^{2} + 81x - 81 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(117253279508625\) \(\medspace = 3^{3}\cdot 5^{3}\cdot 13^{3}\cdot 251^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.52\) | 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: | $3^{1/2}5^{1/2}13^{1/2}251^{1/2}\approx 221.23516899444357$ | ||
Ramified primes: | \(3\), \(5\), \(13\), \(251\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{48945}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a$, $\frac{1}{4714803}a^{9}-\frac{229339}{4714803}a^{8}+\frac{28945}{523867}a^{7}-\frac{73762}{4714803}a^{6}-\frac{2076521}{4714803}a^{5}+\frac{266549}{4714803}a^{4}+\frac{172131}{523867}a^{3}-\frac{129178}{4714803}a^{2}+\frac{39751}{523867}a+\frac{37958}{523867}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{102820}{4714803}a^{9}-\frac{334576}{4714803}a^{8}+\frac{36473}{523867}a^{7}+\frac{337586}{4714803}a^{6}-\frac{2750168}{4714803}a^{5}+\frac{989942}{4714803}a^{4}+\frac{186692}{523867}a^{3}-\frac{6768313}{4714803}a^{2}-\frac{1060248}{523867}a+\frac{556277}{523867}$, $\frac{52390}{4714803}a^{9}-\frac{180565}{4714803}a^{8}+\frac{24622}{1571601}a^{7}+\frac{175679}{4714803}a^{6}-\frac{1142369}{4714803}a^{5}+\frac{827225}{4714803}a^{4}+\frac{813523}{1571601}a^{3}+\frac{1250087}{4714803}a^{2}+\frac{26828}{1571601}a+\frac{20488}{523867}$, $\frac{115841}{4714803}a^{9}-\frac{515795}{4714803}a^{8}+\frac{282968}{1571601}a^{7}+\frac{130795}{4714803}a^{6}-\frac{4878106}{4714803}a^{5}+\frac{7915867}{4714803}a^{4}-\frac{891217}{1571601}a^{3}-\frac{16611587}{4714803}a^{2}+\frac{6824254}{1571601}a-\frac{1294654}{523867}$, $\frac{63121}{4714803}a^{9}-\frac{90208}{4714803}a^{8}-\frac{108386}{1571601}a^{7}+\frac{722561}{4714803}a^{6}-\frac{2130242}{4714803}a^{5}-\frac{720877}{4714803}a^{4}+\frac{761680}{1571601}a^{3}+\frac{1193051}{4714803}a^{2}-\frac{1124044}{1571601}a+\frac{303127}{523867}$, $\frac{355225}{4714803}a^{9}-\frac{1436839}{4714803}a^{8}+\frac{50016}{523867}a^{7}+\frac{4340225}{4714803}a^{6}-\frac{5957678}{4714803}a^{5}-\frac{4091725}{4714803}a^{4}+\frac{1049836}{523867}a^{3}+\frac{3494150}{4714803}a^{2}-\frac{1333744}{523867}a+\frac{865571}{523867}$ | 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: | \( 4126.15651933 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 4126.15651933 \cdot 1}{2\cdot\sqrt{117253279508625}}\cr\approx \mathstrut & 1.18777086463 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.1.48945.1 |
Degree 6 sibling: | 6.2.117253279508625.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.1.48945.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | R | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.6.3.2 | $x^{6} + 338 x^{2} - 24167$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(251\) | $\Q_{251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |