Normalized defining polynomial
\( x^{6} - x^{5} - 342x^{4} - 572x^{3} + 19904x^{2} + 400x - 2848 \)
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: | \(1944559427246513\) \(\medspace = 7^{3}\cdot 11^{3}\cdot 1621^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(353.29\) | 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))
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Galois root discriminant: | $7^{1/2}11^{1/2}1621^{1/2}\approx 353.2944947207641$ | ||
Ramified primes: | \(7\), \(11\), \(1621\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{124817}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{342192}a^{5}-\frac{4161}{114064}a^{4}+\frac{2463}{28516}a^{3}+\frac{6245}{42774}a^{2}+\frac{2671}{14258}a-\frac{6302}{21387}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{8}$, which has order $16$ (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: | $\frac{133}{57032}a^{5}+\frac{409}{28516}a^{4}-\frac{39905}{57032}a^{3}-\frac{177999}{28516}a^{2}+\frac{17766}{7129}a+\frac{13241}{7129}$, $\frac{899}{114064}a^{5}-\frac{1171}{114064}a^{4}-\frac{152743}{57032}a^{3}-\frac{26650}{7129}a^{2}+\frac{1103129}{7129}a-\frac{404296}{7129}$, $\frac{373}{57032}a^{5}-\frac{461}{28516}a^{4}-\frac{132497}{57032}a^{3}-\frac{28579}{28516}a^{2}+\frac{2334779}{14258}a+\frac{452975}{7129}$, $\frac{1027}{114064}a^{5}-\frac{2099}{114064}a^{4}-\frac{171731}{57032}a^{3}-\frac{19065}{7129}a^{2}+\frac{2469119}{14258}a-\frac{626374}{7129}$, $\frac{7219}{114064}a^{5}-\frac{4217}{114064}a^{4}-\frac{309105}{14258}a^{3}-\frac{1278373}{28516}a^{2}+\frac{8865482}{7129}a+\frac{3460705}{7129}$ (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: | \( 76637.131679 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 76637.131679 \cdot 16}{2\cdot\sqrt{1944559427246513}}\cr\approx \mathstrut & 0.88981268536 \end{aligned}\] (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: | \(\Q\) $\times$ 5.5.124817.1 |
Degree 5 sibling: | 5.5.124817.1 |
Degree 10 siblings: | 10.10.1944559427246513.1, 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: | 5.5.124817.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | R | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.6.3.1 | $x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(11\) | 11.6.3.2 | $x^{6} + 37 x^{4} + 18 x^{3} + 367 x^{2} - 558 x + 972$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(1621\) | Deg $6$ | $2$ | $3$ | $3$ |