Normalized defining polynomial
\( x^{6} - x^{5} - 12x^{4} + 13x^{3} + 19x^{2} - 10x - 5 \)
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: | \(3570125\) \(\medspace = 5^{3}\cdot 13^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.36\) | 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: | $5^{1/2}13^{2/3}\approx 12.362716315674692$ | ||
Ramified primes: | \(5\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(65=5\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{65}(16,·)$, $\chi_{65}(1,·)$, $\chi_{65}(9,·)$, $\chi_{65}(29,·)$, $\chi_{65}(14,·)$, $\chi_{65}(61,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{151}a^{5}-\frac{44}{151}a^{4}+\frac{68}{151}a^{3}-\frac{42}{151}a^{2}+\frac{13}{151}a+\frac{35}{151}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{4}{151}a^{5}-\frac{25}{151}a^{4}-\frac{30}{151}a^{3}+\frac{285}{151}a^{2}-\frac{99}{151}a-\frac{313}{151}$, $\frac{4}{151}a^{5}-\frac{25}{151}a^{4}-\frac{30}{151}a^{3}+\frac{285}{151}a^{2}-\frac{250}{151}a-\frac{162}{151}$, $\frac{38}{151}a^{5}-\frac{11}{151}a^{4}-\frac{436}{151}a^{3}+\frac{216}{151}a^{2}+\frac{645}{151}a+\frac{122}{151}$, $\frac{21}{151}a^{5}-\frac{18}{151}a^{4}-\frac{233}{151}a^{3}+\frac{326}{151}a^{2}+\frac{273}{151}a-\frac{322}{151}$, $\frac{10}{151}a^{5}+\frac{13}{151}a^{4}-\frac{75}{151}a^{3}-\frac{118}{151}a^{2}-\frac{21}{151}a+\frac{48}{151}$ | 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: | \( 19.2357657825 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 19.2357657825 \cdot 1}{2\cdot\sqrt{3570125}}\cr\approx \mathstrut & 0.325774995608 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 3.3.169.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{5}) \) $\times$ 3.3.169.1 |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(13\) | 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.65.6t1.b.a | $1$ | $ 5 \cdot 13 $ | 6.6.3570125.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.65.6t1.b.b | $1$ | $ 5 \cdot 13 $ | 6.6.3570125.1 | $C_6$ (as 6T1) | $0$ | $1$ |