Normalized defining polynomial
\( x^{6} - 2x^{5} + 5x^{4} + 6x + 3 \)
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: | \(-5400000\) \(\medspace = -\,2^{6}\cdot 3^{3}\cdot 5^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.25\) | 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: | $2^{3/2}3^{3/4}5^{23/20}\approx 41.03944127612018$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-15}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $a^{5}-3a^{4}+7a^{3}-5a^{2}+a+5$, $a^{5}-2a^{4}+3a^{3}+a^{2}-4a-2$ | 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: | \( 13.8203686028 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 13.8203686028 \cdot 1}{2\cdot\sqrt{5400000}}\cr\approx \mathstrut & 0.737619442889 \end{aligned}\]
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: | \(\Q\) $\times$ 5.3.5400000.1 |
Degree 5 sibling: | 5.3.5400000.1 |
Degree 10 siblings: | 10.0.437400000000000.3, 10.4.437400000000000.1 |
Degree 12 sibling: | deg 12 |
Degree 15 sibling: | deg 15 |
Degree 20 siblings: | deg 20, deg 20, deg 20 |
Degree 24 sibling: | deg 24 |
Degree 30 siblings: | deg 30, deg 30, deg 30 |
Degree 40 sibling: | deg 40 |
Minimal sibling: | 5.3.5400000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.4.3.2 | $x^{4} + 6$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.5.5.3 | $x^{5} + 15 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
4.5400000.10t12.c.a | $4$ | $ 2^{6} \cdot 3^{3} \cdot 5^{5}$ | 6.0.5400000.4 | $\PGL(2,5)$ (as 6T14) | $1$ | $-2$ | |
4.5400000.5t5.d.a | $4$ | $ 2^{6} \cdot 3^{3} \cdot 5^{5}$ | 6.0.5400000.4 | $\PGL(2,5)$ (as 6T14) | $1$ | $2$ | |
5.81000000.10t13.g.a | $5$ | $ 2^{6} \cdot 3^{4} \cdot 5^{6}$ | 6.0.5400000.4 | $\PGL(2,5)$ (as 6T14) | $1$ | $1$ | |
* | 5.5400000.6t14.c.a | $5$ | $ 2^{6} \cdot 3^{3} \cdot 5^{5}$ | 6.0.5400000.4 | $\PGL(2,5)$ (as 6T14) | $1$ | $-1$ |
6.77760000000.20t30.b.a | $6$ | $ 2^{12} \cdot 3^{5} \cdot 5^{7}$ | 6.0.5400000.4 | $\PGL(2,5)$ (as 6T14) | $1$ | $0$ |