Normalized defining polynomial
\( x^{6} - x^{5} - 121x^{4} - 3x^{3} + 3636x^{2} - 81x - 23652 \)
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: | \(15400609258321\) \(\medspace = 7^{4}\cdot 283^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(157.73\) | 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: | $7^{3/4}283^{2/3}\approx 185.50166855022897$ | ||
Ramified primes: | \(7\), \(283\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{4}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{4833}a^{5}-\frac{94}{4833}a^{4}+\frac{566}{4833}a^{3}-\frac{121}{537}a^{2}+\frac{67}{179}a+\frac{31}{179}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$
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: | $\frac{119}{4833}a^{5}-\frac{983}{4833}a^{4}-\frac{7289}{4833}a^{3}+\frac{17842}{1611}a^{2}+\frac{1708}{179}a-\frac{15285}{179}$, $\frac{77}{4833}a^{5}+\frac{280}{4833}a^{4}-\frac{7433}{4833}a^{3}-\frac{12199}{1611}a^{2}+\frac{7064}{537}a+\frac{11695}{179}$, $\frac{175}{4833}a^{5}-\frac{1414}{4833}a^{4}-\frac{11035}{4833}a^{3}+\frac{25438}{1611}a^{2}+\frac{10115}{537}a-\frac{22141}{179}$, $\frac{1036}{1611}a^{5}-\frac{2807}{537}a^{4}-\frac{64826}{1611}a^{3}+\frac{455588}{1611}a^{2}+\frac{158771}{537}a-\frac{372095}{179}$, $\frac{3604}{4833}a^{5}-\frac{28927}{4833}a^{4}-\frac{232183}{4833}a^{3}+\frac{540460}{1611}a^{2}+\frac{186688}{537}a-\frac{448009}{179}$ | 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: | \( 27735.5493081 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 27735.5493081 \cdot 3}{2\cdot\sqrt{15400609258321}}\cr\approx \mathstrut & 0.678483129757 \end{aligned}\]
Galois group
A non-solvable group of order 360 |
The 7 conjugacy class representatives for $A_6$ |
Character table for $A_6$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
Twin sextic algebra: | 6.6.192293689.1 |
Degree 6 sibling: | 6.6.192293689.1 |
Degree 10 sibling: | 10.10.60437550349593857881.1 |
Degree 15 siblings: | 15.15.11621759510846642583679213009.1, deg 15 |
Degree 20 sibling: | deg 20 |
Degree 30 siblings: | deg 30, deg 30 |
Degree 36 sibling: | deg 36 |
Degree 40 sibling: | deg 40 |
Degree 45 sibling: | deg 45 |
Minimal sibling: | 6.6.192293689.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.4.3.1 | $x^{4} + 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(283\) | Deg $3$ | $3$ | $1$ | $2$ | |||
Deg $3$ | $3$ | $1$ | $2$ |