Normalized defining polynomial
\( x^{10} - 2x^{9} - 15x^{8} + 24x^{7} + 69x^{6} - 79x^{5} - 112x^{4} + 82x^{3} + 56x^{2} - 29x - 4 \)
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)
| |
Discriminant: | \(2680810943141213\) \(\medspace = 138917^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(34.90\) | 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: | $138917^{1/2}\approx 372.71570935499886$ | ||
Ramified primes: | \(138917\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{138917}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7023}a^{9}+\frac{935}{7023}a^{8}-\frac{1795}{7023}a^{7}-\frac{3394}{7023}a^{6}+\frac{1310}{7023}a^{5}-\frac{1634}{7023}a^{4}-\frac{52}{2341}a^{3}+\frac{1393}{7023}a^{2}-\frac{327}{2341}a+\frac{787}{7023}$
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: | $\frac{3076}{7023}a^{9}-\frac{3370}{7023}a^{8}-\frac{50503}{7023}a^{7}+\frac{31349}{7023}a^{6}+\frac{258209}{7023}a^{5}-\frac{46877}{7023}a^{4}-\frac{150588}{2341}a^{3}-\frac{48323}{7023}a^{2}+\frac{63985}{2341}a+\frac{25969}{7023}$, $\frac{3076}{7023}a^{9}-\frac{3370}{7023}a^{8}-\frac{50503}{7023}a^{7}+\frac{31349}{7023}a^{6}+\frac{258209}{7023}a^{5}-\frac{46877}{7023}a^{4}-\frac{150588}{2341}a^{3}-\frac{48323}{7023}a^{2}+\frac{66326}{2341}a+\frac{25969}{7023}$, $\frac{2108}{7023}a^{9}-\frac{2483}{7023}a^{8}-\frac{33578}{7023}a^{7}+\frac{22954}{7023}a^{6}+\frac{162970}{7023}a^{5}-\frac{31294}{7023}a^{4}-\frac{86206}{2341}a^{3}-\frac{48331}{7023}a^{2}+\frac{29371}{2341}a+\frac{22637}{7023}$, $\frac{2108}{7023}a^{9}-\frac{2483}{7023}a^{8}-\frac{33578}{7023}a^{7}+\frac{22954}{7023}a^{6}+\frac{162970}{7023}a^{5}-\frac{31294}{7023}a^{4}-\frac{86206}{2341}a^{3}-\frac{48331}{7023}a^{2}+\frac{31712}{2341}a+\frac{22637}{7023}$, $\frac{1949}{7023}a^{9}-\frac{3665}{7023}a^{8}-\frac{29093}{7023}a^{7}+\frac{42898}{7023}a^{6}+\frac{130255}{7023}a^{5}-\frac{136684}{7023}a^{4}-\frac{63892}{2341}a^{3}+\frac{144539}{7023}a^{2}+\frac{25180}{2341}a-\frac{53335}{7023}$, $\frac{682}{7023}a^{9}-\frac{1423}{7023}a^{8}-\frac{9211}{7023}a^{7}+\frac{16928}{7023}a^{6}+\frac{29591}{7023}a^{5}-\frac{53915}{7023}a^{4}+\frac{1992}{2341}a^{3}+\frac{44059}{7023}a^{2}-\frac{12324}{2341}a-\frac{4037}{7023}$, $\frac{299}{7023}a^{9}-\frac{1355}{7023}a^{8}-\frac{2957}{7023}a^{7}+\frac{17575}{7023}a^{6}+\frac{5425}{7023}a^{5}-\frac{67186}{7023}a^{4}-\frac{1502}{2341}a^{3}+\frac{100472}{7023}a^{2}+\frac{2890}{2341}a-\frac{45607}{7023}$, $\frac{1355}{7023}a^{9}-\frac{4238}{7023}a^{8}-\frac{16313}{7023}a^{7}+\frac{50356}{7023}a^{6}+\frac{47392}{7023}a^{5}-\frac{149308}{7023}a^{4}-\frac{4912}{2341}a^{3}+\frac{89627}{7023}a^{2}-\frac{7659}{2341}a-\frac{1111}{7023}$, $\frac{652}{2341}a^{9}-\frac{1381}{2341}a^{8}-\frac{9204}{2341}a^{7}+\frac{15744}{2341}a^{6}+\frac{37111}{2341}a^{5}-\frac{44692}{2341}a^{4}-\frac{43187}{2341}a^{3}+\frac{25679}{2341}a^{2}+\frac{6504}{2341}a+\frac{445}{2341}$ | 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: | \( 53587.2381796 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 53587.2381796 \cdot 1}{2\cdot\sqrt{2680810943141213}}\cr\approx \mathstrut & 0.529905273790 \end{aligned}\]
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: | 5.5.138917.1 |
Degree 6 sibling: | 6.6.2680810943141213.1 |
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: | 5.5.138917.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(138917\) | $\Q_{138917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{138917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{138917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{138917}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |