Normalized defining polynomial
\( x^{10} - 2x^{9} - 2x^{8} + 2x^{7} + 5x^{6} + 6x^{5} - 27x^{4} + 21x^{3} + 25x^{2} - 51x + 14 \)
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: | \(-8776234231883\) \(\medspace = -\,20627^{3}\) | 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: | \(19.69\) | 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: | $20627^{1/2}\approx 143.62102910089456$ | ||
Ramified primes: | \(20627\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-20627}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{15788}a^{9}-\frac{3793}{15788}a^{8}-\frac{3607}{15788}a^{7}+\frac{1731}{15788}a^{6}+\frac{1398}{3947}a^{5}+\frac{2009}{7894}a^{4}+\frac{3155}{15788}a^{3}+\frac{1680}{3947}a^{2}+\frac{6337}{15788}a+\frac{2859}{7894}$
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: | $\frac{609}{7894}a^{9}-\frac{471}{3947}a^{8}-\frac{2131}{7894}a^{7}+\frac{165}{3947}a^{6}+\frac{1607}{3947}a^{5}+\frac{3858}{3947}a^{4}-\frac{12635}{7894}a^{3}+\frac{7335}{7894}a^{2}+\frac{9401}{3947}a-\frac{11337}{3947}$, $\frac{609}{7894}a^{9}-\frac{471}{3947}a^{8}-\frac{2131}{7894}a^{7}+\frac{165}{3947}a^{6}+\frac{1607}{3947}a^{5}+\frac{3858}{3947}a^{4}-\frac{12635}{7894}a^{3}+\frac{7335}{7894}a^{2}+\frac{13348}{3947}a-\frac{11337}{3947}$, $\frac{3425}{15788}a^{9}-\frac{5395}{15788}a^{8}-\frac{7759}{15788}a^{7}+\frac{281}{15788}a^{6}+\frac{4386}{3947}a^{5}+\frac{13045}{7894}a^{4}-\frac{72057}{15788}a^{3}+\frac{18283}{7894}a^{2}+\frac{82559}{15788}a-\frac{51743}{7894}$, $\frac{194}{3947}a^{9}+\frac{547}{7894}a^{8}-\frac{1139}{3947}a^{7}-\frac{3309}{7894}a^{6}-\frac{577}{3947}a^{5}+\frac{5880}{3947}a^{4}+\frac{4232}{3947}a^{3}-\frac{9501}{7894}a^{2}-\frac{225}{7894}a-\frac{3762}{3947}$, $\frac{23}{3947}a^{9}-\frac{405}{3947}a^{8}-\frac{74}{3947}a^{7}+\frac{343}{3947}a^{6}+\frac{2312}{3947}a^{5}+\frac{1633}{3947}a^{4}-\frac{2428}{3947}a^{3}+\frac{627}{3947}a^{2}-\frac{4235}{3947}a+\frac{1263}{3947}$, $\frac{6317}{15788}a^{9}-\frac{9985}{15788}a^{8}-\frac{19123}{15788}a^{7}+\frac{9431}{15788}a^{6}+\frac{9621}{3947}a^{5}+\frac{28877}{7894}a^{4}-\frac{167989}{15788}a^{3}+\frac{10918}{3947}a^{2}+\frac{229281}{15788}a-\frac{127473}{7894}$ | 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: | \( 1078.92283485 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 1078.92283485 \cdot 1}{2\cdot\sqrt{8776234231883}}\cr\approx \mathstrut & 0.722713012022 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_{6}$ |
Character table for $S_{6}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.0.20627.1, 6.4.8776234231883.1 |
Degree 12 siblings: | data not computed |
Degree 15 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.0.20627.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(20627\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |