Normalized defining polynomial
\( x^{6} - x^{5} + 3x^{4} - 3x^{3} + 48x^{2} + 57x + 120 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-898254927\) \(\medspace = -\,3^{4}\cdot 223^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.06\) | 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: | $3^{4/5}223^{1/2}\approx 35.962463598304126$ | ||
Ramified primes: | \(3\), \(223\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-223}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{23}a^{5}-\frac{11}{23}a^{4}-\frac{2}{23}a^{3}-\frac{6}{23}a^{2}-\frac{7}{23}a-\frac{11}{23}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{8}$, which has order $8$
Unit group
Rank: | $2$ | 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: | $\frac{3}{23}a^{5}-\frac{10}{23}a^{4}+\frac{17}{23}a^{3}-\frac{18}{23}a^{2}+\frac{71}{23}a-\frac{79}{23}$, $\frac{2}{23}a^{5}-\frac{22}{23}a^{4}+\frac{19}{23}a^{3}-\frac{12}{23}a^{2}+\frac{193}{23}a-\frac{91}{23}$ | 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: | \( 58.0941834523 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 58.0941834523 \cdot 8}{2\cdot\sqrt{898254927}}\cr\approx \mathstrut & 1.92323539992 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $\PGL(2,5)$ |
Character table for $\PGL(2,5)$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ 5.3.18063.1 |
Degree 5 sibling: | 5.3.18063.1 |
Degree 10 siblings: | 10.4.72758649087.1, 10.0.3618214860447423.1 |
Degree 12 sibling: | deg 12 |
Degree 15 sibling: | deg 15 |
Degree 20 siblings: | deg 20, 20.6.1180522086783240923185887.1, deg 20 |
Degree 24 sibling: | deg 24 |
Degree 30 siblings: | deg 30, deg 30, deg 30 |
Degree 40 sibling: | deg 40 |
Minimal sibling: | 5.3.18063.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 | $[\ ]$ |
3.5.4.1 | $x^{5} + 3$ | $5$ | $1$ | $4$ | $F_5$ | $[\ ]_{5}^{4}$ | |
\(223\) | Deg $6$ | $2$ | $3$ | $3$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.223.2t1.a.a | $1$ | $ 223 $ | \(\Q(\sqrt{-223}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
4.898254927.10t12.b.a | $4$ | $ 3^{4} \cdot 223^{3}$ | 6.0.898254927.1 | $\PGL(2,5)$ (as 6T14) | $1$ | $-2$ | |
4.18063.5t5.b.a | $4$ | $ 3^{4} \cdot 223 $ | 6.0.898254927.1 | $\PGL(2,5)$ (as 6T14) | $1$ | $2$ | |
5.4028049.10t13.b.a | $5$ | $ 3^{4} \cdot 223^{2}$ | 6.0.898254927.1 | $\PGL(2,5)$ (as 6T14) | $1$ | $1$ | |
* | 5.898254927.6t14.b.a | $5$ | $ 3^{4} \cdot 223^{3}$ | 6.0.898254927.1 | $\PGL(2,5)$ (as 6T14) | $1$ | $-1$ |
6.898254927.20t30.b.a | $6$ | $ 3^{4} \cdot 223^{3}$ | 6.0.898254927.1 | $\PGL(2,5)$ (as 6T14) | $1$ | $0$ |