Normalized defining polynomial
\( x^{10} - 3x^{9} - 4x^{8} + 14x^{7} + 10x^{6} - 41x^{5} - 3x^{4} + 58x^{3} - 20x^{2} - 24x + 16 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-3301958349971\) \(\medspace = -\,14891^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.86\) | 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: | $14891^{1/2}\approx 122.02868515230344$ | ||
Ramified primes: | \(14891\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-14891}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{1592}a^{9}+\frac{129}{1592}a^{8}+\frac{77}{398}a^{7}+\frac{37}{796}a^{6}+\frac{113}{796}a^{5}-\frac{457}{1592}a^{4}+\frac{169}{1592}a^{3}-\frac{359}{796}a^{2}-\frac{9}{199}a+\frac{3}{199}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{137}{398}a^{9}-\frac{673}{796}a^{8}-\frac{1377}{796}a^{7}+\frac{691}{199}a^{6}+\frac{2107}{398}a^{5}-\frac{1952}{199}a^{4}-\frac{5235}{796}a^{3}+\frac{9631}{796}a^{2}+\frac{1081}{398}a-\frac{744}{199}$, $\frac{18}{199}a^{9}-\frac{66}{199}a^{8}-\frac{28}{199}a^{7}+\frac{475}{398}a^{6}+\frac{88}{199}a^{5}-\frac{664}{199}a^{4}+\frac{256}{199}a^{3}+\frac{608}{199}a^{2}-\frac{801}{398}a+\frac{34}{199}$, $\frac{305}{1592}a^{9}-\frac{853}{1592}a^{8}-\frac{591}{796}a^{7}+\frac{1733}{796}a^{6}+\frac{1431}{796}a^{5}-\frac{9637}{1592}a^{4}-\frac{593}{1592}a^{3}+\frac{1332}{199}a^{2}-\frac{913}{398}a-\frac{80}{199}$, $\frac{257}{1592}a^{9}-\frac{677}{1592}a^{8}-\frac{421}{796}a^{7}+\frac{1151}{796}a^{6}+\frac{783}{796}a^{5}-\frac{6805}{1592}a^{4}+\frac{847}{1592}a^{3}+\frac{1529}{398}a^{2}-\frac{522}{199}a+\frac{174}{199}$, $\frac{381}{1592}a^{9}-\frac{601}{1592}a^{8}-\frac{1225}{796}a^{7}+\frac{963}{796}a^{6}+\frac{3651}{796}a^{5}-\frac{6161}{1592}a^{4}-\frac{11629}{1592}a^{3}+\frac{1957}{398}a^{2}+\frac{750}{199}a-\frac{449}{199}$, $\frac{305}{1592}a^{9}-\frac{853}{1592}a^{8}-\frac{591}{796}a^{7}+\frac{1733}{796}a^{6}+\frac{1431}{796}a^{5}-\frac{9637}{1592}a^{4}-\frac{593}{1592}a^{3}+\frac{1332}{199}a^{2}-\frac{1311}{398}a+\frac{119}{199}$ | 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: | \( 306.644035797 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 306.644035797 \cdot 1}{2\cdot\sqrt{3301958349971}}\cr\approx \mathstrut & 0.334871601196 \end{aligned}\]
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: | 5.3.14891.1 |
Degree 6 sibling: | 6.0.3301958349971.2 |
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: | 5.3.14891.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(14891\) | $\Q_{14891}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{14891}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{14891}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{14891}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |