Normalized defining polynomial
\( x^{6} - 2x^{5} - 41x^{4} + 66x^{3} + 302x^{2} + 16x - 151 \)
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: | \(8340544000\) \(\medspace = 2^{9}\cdot 5^{3}\cdot 19^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.03\) | 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: | $2^{3/2}5^{1/2}19^{2/3}\approx 45.0331572624958$ | ||
Ramified primes: | \(2\), \(5\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{10}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(760=2^{3}\cdot 5\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{760}(1,·)$, $\chi_{760}(229,·)$, $\chi_{760}(121,·)$, $\chi_{760}(201,·)$, $\chi_{760}(429,·)$, $\chi_{760}(349,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{17279}a^{5}+\frac{2992}{17279}a^{4}+\frac{7485}{17279}a^{3}-\frac{707}{17279}a^{2}-\frac{8418}{17279}a+\frac{6585}{17279}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{46}{17279}a^{5}-\frac{600}{17279}a^{4}-\frac{1270}{17279}a^{3}+\frac{19315}{17279}a^{2}-\frac{7090}{17279}a-\frac{25391}{17279}$, $\frac{202}{17279}a^{5}-\frac{381}{17279}a^{4}-\frac{8582}{17279}a^{3}+\frac{12697}{17279}a^{2}+\frac{62022}{17279}a+\frac{16966}{17279}$, $\frac{202}{17279}a^{5}-\frac{381}{17279}a^{4}-\frac{8582}{17279}a^{3}+\frac{12697}{17279}a^{2}+\frac{79301}{17279}a-\frac{52150}{17279}$, $\frac{214}{17279}a^{5}+\frac{965}{17279}a^{4}-\frac{5157}{17279}a^{3}-\frac{30345}{17279}a^{2}-\frac{21715}{17279}a+\frac{26870}{17279}$, $\frac{1633}{17279}a^{5}-\frac{4021}{17279}a^{4}-\frac{62364}{17279}a^{3}+\frac{141394}{17279}a^{2}+\frac{335791}{17279}a-\frac{236139}{17279}$ | 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: | \( 380.611218272 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{0}\cdot 380.611218272 \cdot 2}{2\cdot\sqrt{8340544000}}\cr\approx \mathstrut & 0.266725396732 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{10}) \), 3.3.361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
Twin sextic algebra: | \(\Q\) $\times$ \(\Q(\sqrt{10}) \) $\times$ 3.3.361.1 |
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 | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | R | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.6.9.3 | $x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
\(5\) | 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
\(19\) | 19.6.4.3 | $x^{6} + 54 x^{5} + 1168 x^{4} + 12926 x^{3} + 104347 x^{2} + 738556 x + 220465$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.40.2t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.760.6t1.a.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.6.8340544000.1 | $C_6$ (as 6T1) | $0$ | $1$ |
* | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.760.6t1.a.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.6.8340544000.1 | $C_6$ (as 6T1) | $0$ | $1$ |