Normalized defining polynomial
\( x^{10} - 2x^{9} - 15x^{8} + 28x^{7} + 61x^{6} - 107x^{5} - 72x^{4} + 110x^{3} + 36x^{2} - 25x - 8 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1852754544393661\) \(\medspace = 263^{3}\cdot 467^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.63\) | 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: | $263^{1/2}467^{1/2}\approx 350.4582714104491$ | ||
Ramified primes: | \(263\), \(467\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{122821}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6631}a^{9}-\frac{996}{6631}a^{8}+\frac{1990}{6631}a^{7}-\frac{1994}{6631}a^{6}-\frac{572}{6631}a^{5}-\frac{95}{349}a^{4}-\frac{2903}{6631}a^{3}+\frac{1207}{6631}a^{2}+\frac{489}{6631}a-\frac{2028}{6631}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | 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: | $\frac{2284}{6631}a^{9}-\frac{7062}{6631}a^{8}-\frac{30230}{6631}a^{7}+\frac{100666}{6631}a^{6}+\frac{86062}{6631}a^{5}-\frac{20144}{349}a^{4}+\frac{20441}{6631}a^{3}+\frac{362997}{6631}a^{2}-\frac{70073}{6631}a-\frac{63193}{6631}$, $\frac{1659}{6631}a^{9}-\frac{1245}{6631}a^{8}-\frac{27352}{6631}a^{7}+\frac{14085}{6631}a^{6}+\frac{131905}{6631}a^{5}-\frac{1951}{349}a^{4}-\frac{214163}{6631}a^{3}-\frac{149}{6631}a^{2}+\frac{101734}{6631}a+\frac{23989}{6631}$, $\frac{4364}{6631}a^{9}-\frac{9870}{6631}a^{8}-\frac{61929}{6631}a^{7}+\frac{137307}{6631}a^{6}+\frac{215871}{6631}a^{5}-\frac{26841}{349}a^{4}-\frac{122840}{6631}a^{3}+\frac{473135}{6631}a^{2}-\frac{21079}{6631}a-\frac{90641}{6631}$, $\frac{1129}{6631}a^{9}-\frac{3845}{6631}a^{8}-\frac{14461}{6631}a^{7}+\frac{56362}{6631}a^{6}+\frac{37205}{6631}a^{5}-\frac{11978}{349}a^{4}+\frac{18120}{6631}a^{3}+\frac{248695}{6631}a^{2}-\frac{24816}{6631}a-\frac{54965}{6631}$, $\frac{3840}{6631}a^{9}-\frac{11815}{6631}a^{8}-\frac{50360}{6631}a^{7}+\frac{167620}{6631}a^{6}+\frac{137632}{6631}a^{5}-\frac{33250}{349}a^{4}+\frac{58870}{6631}a^{3}+\frac{589970}{6631}a^{2}-\frac{111540}{6631}a-\frac{102191}{6631}$, $\frac{3222}{6631}a^{9}-\frac{6339}{6631}a^{8}-\frac{46814}{6631}a^{7}+\frac{86974}{6631}a^{6}+\frac{172840}{6631}a^{5}-\frac{16769}{349}a^{4}-\frac{129745}{6631}a^{3}+\frac{288321}{6631}a^{2}-\frac{15882}{6631}a-\frac{42467}{6631}$, $\frac{414}{6631}a^{9}-\frac{1222}{6631}a^{8}-\frac{5015}{6631}a^{7}+\frac{16621}{6631}a^{6}+\frac{8539}{6631}a^{5}-\frac{3034}{349}a^{4}+\frac{31524}{6631}a^{3}+\frac{42159}{6631}a^{2}-\frac{42901}{6631}a+\frac{2545}{6631}$, $\frac{1870}{6631}a^{9}-\frac{5840}{6631}a^{8}-\frac{25215}{6631}a^{7}+\frac{84045}{6631}a^{6}+\frac{77523}{6631}a^{5}-\frac{17110}{349}a^{4}-\frac{11083}{6631}a^{3}+\frac{320838}{6631}a^{2}-\frac{33803}{6631}a-\frac{59107}{6631}$, $\frac{938}{6631}a^{9}+\frac{723}{6631}a^{8}-\frac{16584}{6631}a^{7}-\frac{13692}{6631}a^{6}+\frac{86778}{6631}a^{5}+\frac{3375}{349}a^{4}-\frac{150186}{6631}a^{3}-\frac{74676}{6631}a^{2}+\frac{54191}{6631}a+\frac{14095}{6631}$ | 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: | \( 41956.5931703 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 41956.5931703 \cdot 1}{2\cdot\sqrt{1852754544393661}}\cr\approx \mathstrut & 0.499069735330 \end{aligned}\]
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: | 5.5.122821.1 |
Degree 6 sibling: | 6.6.1852754544393661.1 |
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: | 5.5.122821.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(263\) | $\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
\(467\) | $\Q_{467}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |