Normalized defining polynomial
\( x^{10} - x^{9} - 4x^{8} + 9x^{7} + 10x^{6} - 25x^{5} - 6x^{4} + 13x^{3} + 28x^{2} - 33x + 9 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-260807798403\) \(\medspace = -\,3^{5}\cdot 181^{4}\) | 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: | \(13.86\) | 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))
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Galois root discriminant: | $3^{1/2}181^{2/3}\approx 55.4216788307774$ | ||
Ramified primes: | \(3\), \(181\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{1770}a^{9}+\frac{23}{1770}a^{8}-\frac{337}{1770}a^{7}-\frac{19}{295}a^{6}+\frac{407}{885}a^{5}+\frac{41}{1770}a^{4}+\frac{31}{590}a^{3}+\frac{95}{354}a^{2}+\frac{404}{885}a+\frac{129}{295}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1814}{885} a^{9} - \frac{758}{885} a^{8} - \frac{7748}{885} a^{7} + \frac{3933}{295} a^{6} + \frac{25196}{885} a^{5} - \frac{30941}{885} a^{4} - \frac{9846}{295} a^{3} + \frac{1348}{177} a^{2} + \frac{55022}{885} a - \frac{9003}{295} \) (order $6$) | 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: | $\frac{100}{59}a^{9}-\frac{61}{118}a^{8}-\frac{424}{59}a^{7}+\frac{1213}{118}a^{6}+\frac{2851}{118}a^{5}-\frac{3069}{118}a^{4}-\frac{1674}{59}a^{3}+\frac{305}{118}a^{2}+\frac{2802}{59}a-\frac{1306}{59}$, $\frac{758}{885}a^{9}-\frac{266}{885}a^{8}-\frac{3221}{885}a^{7}+\frac{1581}{295}a^{6}+\frac{10787}{885}a^{5}-\frac{12287}{885}a^{4}-\frac{4232}{295}a^{3}+\frac{502}{177}a^{2}+\frac{23054}{885}a-\frac{3856}{295}$, $\frac{949}{1770}a^{9}-\frac{149}{885}a^{8}-\frac{1934}{885}a^{7}+\frac{1993}{590}a^{6}+\frac{6578}{885}a^{5}-\frac{14191}{1770}a^{4}-\frac{2253}{295}a^{3}+\frac{385}{177}a^{2}+\frac{27817}{1770}a-\frac{2364}{295}$, $\frac{1187}{590}a^{9}-\frac{429}{590}a^{8}-\frac{2507}{295}a^{7}+\frac{3731}{295}a^{6}+\frac{16613}{590}a^{5}-\frac{9444}{295}a^{4}-\frac{9557}{295}a^{3}+\frac{547}{118}a^{2}+\frac{34861}{590}a-\frac{8501}{295}$ | 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: | \( 107.122533035 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 107.122533035 \cdot 2}{6\cdot\sqrt{260807798403}}\cr\approx \mathstrut & 0.684697329565 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.3.98283.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 5 sibling: | 5.3.98283.1 |
Degree 6 sibling: | 6.0.28978644267.3 |
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: | 5.3.98283.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(181\) | $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
181.3.2.3 | $x^{3} + 543$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
181.3.2.3 | $x^{3} + 543$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |