Normalized defining polynomial
\( x^{6} - 2x^{5} - x^{4} - 18x^{3} - 211x^{2} - 415x - 538 \)
Invariants
Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1278983549\) \(\medspace = 29^{3}\cdot 229^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(32.95\) | 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: | $29^{1/2}229^{1/2}\approx 81.49233092751734$ | ||
Ramified primes: | \(29\), \(229\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{158033}a^{5}-\frac{34718}{158033}a^{4}-\frac{47604}{158033}a^{3}+\frac{69365}{158033}a^{2}+\frac{1361}{6871}a+\frac{77578}{158033}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{8}$, which has order $8$
Unit group
Rank: | $3$ | 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: | $\frac{478}{158033}a^{5}-\frac{1739}{158033}a^{4}+\frac{2040}{158033}a^{3}-\frac{30460}{158033}a^{2}-\frac{2187}{6871}a-\frac{55471}{158033}$, $\frac{478}{158033}a^{5}-\frac{1739}{158033}a^{4}+\frac{2040}{158033}a^{3}-\frac{30460}{158033}a^{2}-\frac{2187}{6871}a-\frac{371537}{158033}$, $\frac{15861}{158033}a^{5}-\frac{75226}{158033}a^{4}+\frac{192663}{158033}a^{3}-\frac{817646}{158033}a^{2}-\frac{49958}{6871}a-\frac{3298973}{158033}$ | 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: | \( 46.9529526652 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 46.9529526652 \cdot 8}{2\cdot\sqrt{1278983549}}\cr\approx \mathstrut & 0.829297103504 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 6T11):
A solvable group of order 48 |
The 10 conjugacy class representatives for $S_4\times C_2$ |
Character table for $S_4\times C_2$ |
Intermediate fields
3.3.229.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 4.0.229.1 $\times$ \(\Q(\sqrt{6641}) \) |
Degree 6 sibling: | 6.2.292887232721.1 |
Degree 8 siblings: | 8.0.1945064112500161.2, 8.0.37090522921.1 |
Degree 12 siblings: | deg 12, deg 12, deg 12, deg 12, deg 12, deg 12 |
Degree 16 sibling: | deg 16 |
Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(29\) | 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(229\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.6641.2t1.a.a | $1$ | $ 29 \cdot 229 $ | \(\Q(\sqrt{6641}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.229.2t1.a.a | $1$ | $ 229 $ | \(\Q(\sqrt{229}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
2.192589.6t3.b.a | $2$ | $ 29^{2} \cdot 229 $ | 6.6.292887232721.1 | $D_{6}$ (as 6T3) | $1$ | $2$ | |
* | 2.229.3t2.a.a | $2$ | $ 229 $ | 3.3.229.1 | $S_3$ (as 3T2) | $1$ | $2$ |
3.229.4t5.a.a | $3$ | $ 229 $ | 4.0.229.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.5585081.6t11.a.a | $3$ | $ 29^{3} \cdot 229 $ | 6.2.1278983549.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
3.1278983549.6t11.a.a | $3$ | $ 29^{3} \cdot 229^{2}$ | 6.2.1278983549.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.52441.6t8.c.a | $3$ | $ 229^{2}$ | 4.0.229.1 | $S_4$ (as 4T5) | $1$ | $-1$ |