Normalized defining polynomial
\( x^{6} - x^{5} - 242x^{4} + 326x^{3} + 11662x^{2} - 16095x - 45463 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[6, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(6438696919952653\) \(\medspace = 186037^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(431.32\) | 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: | $186037^{1/2}\approx 431.3200667717652$ | ||
Ramified primes: | \(186037\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{186037}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{26830996}a^{5}-\frac{5394545}{13415498}a^{4}+\frac{150124}{6707749}a^{3}-\frac{2338343}{13415498}a^{2}-\frac{2255765}{6707749}a-\frac{10269599}{26830996}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
Rank: | $5$ | 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: | $\frac{1244823}{13415498}a^{5}-\frac{5427153}{6707749}a^{4}-\frac{107482160}{6707749}a^{3}+\frac{1037523607}{6707749}a^{2}-\frac{992913541}{6707749}a-\frac{6241965877}{13415498}$, $\frac{2869465}{13415498}a^{5}+\frac{883377}{6707749}a^{4}-\frac{327541690}{6707749}a^{3}-\frac{166226775}{6707749}a^{2}+\frac{12962491909}{6707749}a+\frac{36127884913}{13415498}$, $\frac{412557}{13415498}a^{5}+\frac{31396}{6707749}a^{4}-\frac{49540890}{6707749}a^{3}+\frac{8688129}{6707749}a^{2}+\frac{2350622460}{6707749}a+\frac{597412641}{13415498}$, $\frac{1528025}{13415498}a^{5}-\frac{1155752}{6707749}a^{4}-\frac{164444129}{6707749}a^{3}+\frac{218283968}{6707749}a^{2}+\frac{5001401278}{6707749}a+\frac{14392736465}{13415498}$, $\frac{346358}{6707749}a^{5}+\frac{26164676}{6707749}a^{4}+\frac{73205564}{6707749}a^{3}-\frac{2770558158}{6707749}a^{2}+\frac{3305241367}{6707749}a+\frac{10639028196}{6707749}$ (assuming GRH) | 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: | \( 1073944.18308 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 1073944.18308 \cdot 4}{2\cdot\sqrt{6438696919952653}}\cr\approx \mathstrut & 1.71313934066 \end{aligned}\] (assuming GRH)
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: | \(\Q\) $\times$ 5.5.186037.1 |
Degree 5 sibling: | 5.5.186037.1 |
Degree 10 siblings: | 10.10.6438696919952653.1, deg 10 |
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: | 5.5.186037.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(186037\) | Deg $6$ | $2$ | $3$ | $3$ |