Normalized defining polynomial
\( x^{6} - x^{5} - 31x^{4} + 4x^{3} + 162x^{2} - 81x - 27 \)
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: | \(891474493\) \(\medspace = 7^{4}\cdot 13^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(31.02\) | 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: | $7^{2/3}13^{5/6}\approx 31.02307339173177$ | ||
Ramified primes: | \(7\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(91=7\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(81,·)$, $\chi_{91}(88,·)$, $\chi_{91}(9,·)$, $\chi_{91}(30,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{9}a^{2}+\frac{4}{9}a-\frac{1}{3}$, $\frac{1}{243}a^{5}-\frac{7}{243}a^{4}+\frac{11}{243}a^{3}-\frac{62}{243}a^{2}+\frac{16}{81}a+\frac{13}{27}$
Monogenic: | No | |
Index: | $27$ | |
Inessential primes: | $3$ |
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: | $\frac{4}{243}a^{5}-\frac{1}{243}a^{4}-\frac{145}{243}a^{3}-\frac{32}{243}a^{2}+\frac{316}{81}a+\frac{16}{27}$, $\frac{1}{81}a^{5}+\frac{2}{81}a^{4}-\frac{25}{81}a^{3}-\frac{71}{81}a^{2}+\frac{10}{27}a+\frac{1}{9}$, $\frac{1}{9}a^{5}-\frac{32}{9}a^{3}-\frac{8}{3}a^{2}+\frac{157}{9}a+\frac{5}{3}$, $\frac{2}{81}a^{5}-\frac{5}{81}a^{4}-\frac{41}{81}a^{3}+\frac{29}{81}a^{2}+\frac{35}{27}a+\frac{5}{9}$, $\frac{4}{243}a^{5}-\frac{28}{243}a^{4}-\frac{37}{243}a^{3}+\frac{481}{243}a^{2}-\frac{152}{81}a-\frac{56}{27}$ | 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: | \( 549.032432659 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 549.032432659 \cdot 3}{2\cdot\sqrt{891474493}}\cr\approx \mathstrut & 1.76528476153 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{13}) \), 3.3.8281.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | 3.3.8281.2 $\times$ \(\Q(\sqrt{13}) \) $\times$ \(\Q\) |
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 | ${\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.1.0.1}{1} }^{6}$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.6.4.1 | $x^{6} + 14 x^{3} - 245$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
\(13\) | 13.6.5.2 | $x^{6} + 13$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.91.3t1.a.a | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.91.6t1.i.a | $1$ | $ 7 \cdot 13 $ | 6.6.891474493.2 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.91.3t1.a.b | $1$ | $ 7 \cdot 13 $ | 3.3.8281.2 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.91.6t1.i.b | $1$ | $ 7 \cdot 13 $ | 6.6.891474493.2 | $C_6$ (as 6T1) | $0$ | $1$ |