Normalized defining polynomial
\( x^{6} + 65x^{4} + 650x^{2} + 1625 \)
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: |
\(-2970344000\)
\(\medspace = -\,2^{6}\cdot 5^{3}\cdot 13^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(37.91\) | 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: | $2\cdot 5^{1/2}13^{5/6}\approx 37.914132609839655$ | ||
Ramified primes: |
\(2\), \(5\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-65}) \) | ||
$\card{ \Gal(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(260=2^{2}\cdot 5\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{260}(1,·)$, $\chi_{260}(259,·)$, $\chi_{260}(179,·)$, $\chi_{260}(81,·)$, $\chi_{260}(199,·)$, $\chi_{260}(61,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-65}) \), 6.0.2970344000.1$^{3}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{125}a^{4}+\frac{1}{5}$, $\frac{1}{125}a^{5}+\frac{1}{5}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $32$
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: |
$\frac{2}{125}a^{4}+a^{2}+\frac{32}{5}$, $\frac{1}{125}a^{4}+\frac{2}{5}a^{2}+\frac{6}{5}$
| 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: | \( 5.46019947038 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 5.46019947038 \cdot 32}{2\cdot\sqrt{2970344000}}\cr\approx \mathstrut & 0.397616796493 \end{aligned}\]
Galois group
A cyclic group of order 6 |
The 6 conjugacy class representatives for $C_6$ |
Character table for $C_6$ |
Intermediate fields
\(\Q(\sqrt{-65}) \), 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: | 3.3.169.1 $\times$ \(\Q(\sqrt{-65}) \) $\times$ \(\Q\) |
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 | R | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(13\)
| 13.6.5.5 | $x^{6} + 65$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.260.2t1.a.a | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | \(\Q(\sqrt{-65}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.13.3t1.a.a | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.260.6t1.b.a | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | 6.0.2970344000.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
* | 1.13.3t1.a.b | $1$ | $ 13 $ | 3.3.169.1 | $C_3$ (as 3T1) | $0$ | $1$ |
* | 1.260.6t1.b.b | $1$ | $ 2^{2} \cdot 5 \cdot 13 $ | 6.0.2970344000.1 | $C_6$ (as 6T1) | $0$ | $-1$ |