Normalized defining polynomial
\( x^{6} - 2x^{5} - 35x^{4} + 50x^{3} + 432x^{2} - 252x + 49 \)
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: | \(-41953352192\) \(\medspace = -\,2^{9}\cdot 19^{2}\cdot 61^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(58.95\) | 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: | $2^{3/2}19^{2/3}61^{1/2}\approx 157.2940561081374$ | ||
Ramified primes: | \(2\), \(19\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-122}) \) | ||
$\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{14511}a^{5}+\frac{4793}{14511}a^{4}-\frac{144}{691}a^{3}-\frac{3541}{14511}a^{2}-\frac{793}{14511}a-\frac{115}{2073}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{10}$, which has order $10$
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: | $\frac{1053197}{14511}a^{5}-\frac{1738691}{14511}a^{4}-\frac{1785923}{691}a^{3}+\frac{39614986}{14511}a^{2}+\frac{470079229}{14511}a-\frac{14615207}{2073}$, $\frac{42319}{14511}a^{5}-\frac{275500}{14511}a^{4}-\frac{13827}{691}a^{3}+\frac{4008554}{14511}a^{2}-\frac{1924987}{14511}a+\frac{48398}{2073}$ | 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: | \( 347.188453338 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 347.188453338 \cdot 10}{2\cdot\sqrt{41953352192}}\cr\approx \mathstrut & 2.10228479190 \end{aligned}\]
Galois group
$C_3\times S_3$ (as 6T5):
A solvable group of order 18 |
The 9 conjugacy class representatives for $S_3\times C_3$ |
Character table for $S_3\times C_3$ |
Intermediate fields
\(\Q(\sqrt{-122}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Galois closure: | deg 18 |
Twin sextic algebra: | 3.1.176168.1 $\times$ 3.3.361.1 |
Degree 9 sibling: | 9.3.5467402811013632.1 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{3}$ | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | R | ${\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.9.7 | $x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
\(19\) | 19.3.2.3 | $x^{3} + 38$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(61\) | 61.6.3.2 | $x^{6} + 26047 x^{2} - 13391879$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |