Normalized defining polynomial
\( x^{10} - 2x^{9} + 6x^{7} - 15x^{6} + 17x^{5} - 23x^{4} - 34x^{3} - 20x^{2} - 64x - 33 \)
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)
| |
Discriminant: | \(-12064136250375\) \(\medspace = -\,3^{3}\cdot 5^{3}\cdot 11^{3}\cdot 139^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(20.33\) | 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: | $3^{1/2}5^{1/2}11^{1/2}139^{1/2}\approx 151.44305860619693$ | ||
Ramified primes: | \(3\), \(5\), \(11\), \(139\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-22935}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2253719}a^{9}-\frac{634074}{2253719}a^{8}+\frac{875761}{2253719}a^{7}+\frac{549343}{2253719}a^{6}+\frac{525334}{2253719}a^{5}+\frac{88169}{2253719}a^{4}+\frac{259323}{2253719}a^{3}-\frac{368769}{2253719}a^{2}+\frac{497379}{2253719}a+\frac{70913}{2253719}$
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: | $\frac{383057}{2253719}a^{9}-\frac{933869}{2253719}a^{8}+\frac{308227}{2253719}a^{7}+\frac{2192240}{2253719}a^{6}-\frac{6210910}{2253719}a^{5}+\frac{8534575}{2253719}a^{4}-\frac{12950547}{2253719}a^{3}-\frac{5454789}{2253719}a^{2}-\frac{4896657}{2253719}a-\frac{16130099}{2253719}$, $\frac{12952}{2253719}a^{9}+\frac{25588}{2253719}a^{8}-\frac{111255}{2253719}a^{7}+\frac{99653}{2253719}a^{6}+\frac{148307}{2253719}a^{5}-\frac{670645}{2253719}a^{4}+\frac{710186}{2253719}a^{3}-\frac{665527}{2253719}a^{2}-\frac{1329813}{2253719}a-\frac{1052176}{2253719}$, $\frac{21998}{2253719}a^{9}-\frac{92961}{2253719}a^{8}+\frac{200466}{2253719}a^{7}+\frac{6036}{2253719}a^{6}-\frac{773700}{2253719}a^{5}+\frac{1343322}{2253719}a^{4}-\frac{1829154}{2253719}a^{3}+\frac{1207938}{2253719}a^{2}-\frac{462503}{2253719}a-\frac{1883093}{2253719}$, $\frac{302071}{2253719}a^{9}-\frac{804320}{2253719}a^{8}+\frac{464811}{2253719}a^{7}+\frac{1513102}{2253719}a^{6}-\frac{5202952}{2253719}a^{5}+\frac{7861733}{2253719}a^{4}-\frac{12075664}{2253719}a^{3}+\frac{148414}{2253719}a^{2}-\frac{7166383}{2253719}a-\frac{9853148}{2253719}$, $\frac{25588}{2253719}a^{9}-\frac{162431}{2253719}a^{8}+\frac{244451}{2253719}a^{7}+\frac{143281}{2253719}a^{6}-\frac{1187443}{2253719}a^{5}+\frac{2349372}{2253719}a^{4}-\frac{1645531}{2253719}a^{3}+\frac{260281}{2253719}a^{2}+\frac{2436378}{2253719}a-\frac{1975670}{2253719}$, $\frac{125586}{2253719}a^{9}-\frac{163937}{2253719}a^{8}-\frac{419973}{2253719}a^{7}+\frac{1197689}{2253719}a^{6}-\frac{774282}{2253719}a^{5}-\frac{1983132}{2253719}a^{4}+\frac{1098728}{2253719}a^{3}-\frac{2805622}{2253719}a^{2}-\frac{6997867}{2253719}a-\frac{3271189}{2253719}$ | 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: | \( 774.89987679 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 774.89987679 \cdot 1}{2\cdot\sqrt{12064136250375}}\cr\approx \mathstrut & 0.44271788516 \end{aligned}\]
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: | 5.3.22935.1 |
Degree 6 sibling: | 6.0.12064136250375.1 |
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: | 5.3.22935.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | R | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(139\) | $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
139.3.0.1 | $x^{3} + 6 x + 137$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
139.6.3.1 | $x^{6} + 115926 x^{2} - 367929803$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |